Talk:Matrix (mathematics)/Archive 2

3704:

transformations, then this should be done, but if knowing about linear transformations is a prerequisite for understanding the use of matrices, then the lead should link to the notion and assume the reader understands it. I cringe at sentences that mention for instance "entries in a field" and then haste to add "like the real numbers" as if that really helps the readers put off by the term; "generalizations of linear functions such as f(x) = 4x" suffers from the same problems (one who is unfamiliar with linear transformations is unlikely to understand what kind of generalizations these could be, and probably just feels even less comfortable). To add something constructive, I think one could start (after the initial definition) by saying something like that matrices serve to treat the set of entries they contain as a unit, when these entries are used together to express linear relations between sets of values. A first example could be the collection of coefficients in (the left hand side of) a system of linear equations (after it has been brought into a standard form). This is much less abstract than talk about linear transformations; people can immediately spot why a rectangular set of coefficients is involved. One could then add that "more generally matrices provide a concrete representation of linear transformations" or something similar. Then one could add that operations can be defined on matrices depending on their interpretation (for instance in systems of equations it is natural to consider interchange of equations, or addition of one equation to another), and that notably composition of linear transformation gives rise to matrix multiplication. But in fact I'm more thinking about an introductory section than about the lead here; the lead should really be in a summary style and not involve detailed examples. Formulating a good lead is a difficult balancing act, and I fully agree that we are putting too much effort into doing that rather than improving the article first.

2276:(Hm... the text above this reply has just changed (edit conflict); curious.) Even if there might be some form of formal justification, calling a 1x1 matrix a scalar would be quite confusing in the context of this article. In the context of matrix algebra, one distinguishes scalar multiplication and matrix multiplication as different operations, and the scalar argument in scalar multiplication is a scalar, not a matrix (seems obvious). If one does want to view scalar multiplication as a form of matrix multiplication, then one needs to represent the scalar by the corresponding multiple of the identity matrix, not by a 1x1 matrix. Of course sufficiently knowledgeable people will know that there is some way (the Kronecker product) to view scalar multiplication as multiplication by a 1x1 matrix, but really, these people do not need to read this article to find that out. IMHO the lead should be kept at a quite basic/broad audience level. 3155:. The function that is involved here is the one from real numbers to real numbers that (assuming the usual convention for variable names) doubles its argument. Now saying that matrices play the same role as functions is still unclear to me. Matrices are used to represent linear transformations, so functions supposedly similarly represent something, but what? The only thing I can think of here is some real-world relation between values that is modeled by the function; however their relation is rather different than linear transformation–matrix. My guess is that you meant to say that matrices play a role relative to linear transformations in linear algebra that is similar to that of expressions relative to functions in (high school) algebra. You may ignore the "Of course" sentence above, which was just a wrong guess about the suggested relation between matrices and functions, but in case you wonder, the explanation is this: a 3093:

algebra functions certainly play a major role. The article as it stood gave no indication what matrices were used for. They are most often used to represent linear functions acting on vectors, analogous to y = 2x, a linear function acting on real numbers. Remember, we are writing for a reader who has little or no mathematics beyond what is taught in secondary school. Mathematicians already know what a matrix is. The fact that most functions cannot be written down is a technical abstraction far beyond the subject matter of this article. If you really want to go there, most matrices cannot be written down, because most real numbers cannot be written down. When I get to your sentence that begins "Of course matrices..." I cannot make heads or tails of it. You seem to confuse a vector space (the codomain of a linear transformation) with the field of scalars (some set where the entires live).

3077:

to ask first what on earth this statement was intended to convey. What particular role do functions play in algebra? Unless they are some kind of morphisms, algebra deals very little with functions, much less then analysis does in any case. And how does this role compare with that of matrices? The main virtue of matrices it that they can be written down, which functions cannot in general. Of course matrices, like many other objects in mathematics, can be defined as functions (from a rectangular index set to some set where the entries live), but this does not seem to be what the sentence is pointing to. There are some other similar interpretations possible of this sentence, but none that is in any way convincing. Please clarify.

1209:. But I concur with your suspicions about the word; Webster's also gives as a main usage 2a as in "mold" (die, e.g. used for casting), so perhaps the (square) matrix casts out a determinant, perhaps not. Unfortunately we should resort to the Oxford Dictionary of the English Language (not available on line unless you subscribe for about $ 500 per year). My skepticism is more related to the claim of "first usage", which as you point out may be inherited lack-of-"wisdom". The claim may be true, but only deep academic-grade research into the first usage would reveal this together with a history of the notion of a "5a: a rectangular array of mathematical elements ". Surely such things were called 2488:

made the minimum patching to avoid major dissonance with my experience (applications in published research, undergraduate and graduate course) of working with matrices for over 60 years -- a major newspaper article in the Guardian Weekly last week indicated a concern about the dearth of academics who contribute in their field of expertize -- but I have seen that other professionals avoid stating their credentials so I say no more on this. However, the concensus in the ratings seems to be that this is a good article, so I may be quite wrong about what it is intended to do. If so, please revert what I have done. If anyone wants a clarification of changes I made, I will be responsive. Thanks.

3272:

understand that we can't explain everything in the lede, but we need to give some idea of what matrices are good for. The sentence "matrices represent functions" is mathematically correct, and focuses on the first application of matrices most students see. We could say exactly that, but it seems to me better to say something that tells the lay reader that matrices are similar to the functions they learned about in high school. How about this: "Matrices are used to represent functions, similar to the functions taught in secondary school, functions which are important in many fields, functions which are most easily expressed in matrix form."

2448:

containing "matrix" & "entry". Here, the search is confused by the many topics that place these words close together (e.g. pieces of solid in which other solids are embedded). Searches on the word pairs "matrix element" and "matrix entry" found 36 and 3 papers respectively. Another search found several papers containing "matrix entry" in the journal "Linear algebra and its applications". A direct search of this journal, that has been publishing since the 1960s, reported 4000 occurrences of "matrix entry" and 5000 of "matrix element". I have tried to accommodate these findings in a small edit. I hope it does not tread on toes.

3619:"The lead section should briefly summarize the most important points covered in an article in such a way that it can stand on its own as a concise version of the article. The reason for the topic being noteworthy should be established early on in the lead. It is even more important here than for the rest of the article that the text be accessible. Consideration should be given to creating interest in reading the whole article. (See news style and summary style.) This allows editors to avoid lengthy paragraphs and over-specific descriptions, because the reader will know that greater detail is saved for the body of the article." 3654:

justify confusing almost all readers from the outset. It may be that 137 underestimates just how forbidding most people find mathematics. I was recently asked, by an accountant!, what mathematics is good for. I started on some high-falutin' explanation when she interupted, "I want you to know I don't think all those little x's and y's really mean anything." This is easy to make fun of, but the person is intelligent, highly educated, and a well-paid professional. Maybe no Knowledge article on mathematics will reach the average reader, but it would be nice to reach at least some readers.

3650:"I can understand Rick Norwood's objection to the sentence "A major application of matrices is to represent linear transformations of vectors in linear algebra", because not everybody will understand it. But this is a serious encyclopedia, not a friendly textbook written for high-school students. I think this sentence gets to the core of what matrices are, in a concise and (compared to most mathematics articles) clear way, and so it should be one of the first sentences of the article. 137.82.175.12 (talk) 22:06, 30 May 2011 (UTC)" 5384:

sufficiently technical manner for a mathematician. I expected that reaction from such readers, and hoped the list of questions that a mathematical reader would naturally raise would assuage these concerns without bogging matters down in detail that can be found elsewhere. This approach is perhaps unusual on WP, especially in math articles that tend to be framed in jargon and without reference to background material for the uninitiated. However, I do not think there is anything confusing here. I await your detailed discussion.

1587:

transformations and systems of linear equations, amongst other things). All of this is covered at a summary level in the first two or three paragraphs of the lead section. If we omit what they look like and how they are used, then we are left with a very abstract definition of matrices purely in terms of their algebraic behaviour. This is possible, but it requires a certain level of mathematical maturity to appreciate and understand, so it would reduce the accessibility of the article for a general reader.

2601:

distribution, commutation (and non-commutation), association, zero matrix, identity matrix, trace, transposed, singular, reciprocal, associate, symmetric, orthogonal, real, Hermitian, unitary matrices, vector space, linear transformations, equivalent matrices, bilinear and quadratic forms, characteristic equation, reduction to diagonal form, eigenvalues, eigenvectors, congruent transformations, orthogonal transformations (Margenau and Murphy, Mathematics of Physics and Chemistry).

6248:@Cerniagigante You also learned nothing from the numerical support given in my argument. Had you applied a similar googlebooks/googlescholar search technique, you would find that "formulae" has 61% of the usage of "formulas" on googlebooks, and 66% of the usage of "formulas" on googlescholar. So, that indicates two things: A. "formulas" is more preferable, and B. "formulae" has strong usage, so argument to replace all occurences of "formulas" with "formulae" is extremely weak. 1606:

transformations this way. Neat." Why do you need matrices to do linear transformations? If you don't, then what are matrices absolutely necessary for? I'm not suggesting the first sentence answer all of these, I'm just trying to get at how to think about introducing the idea of matrices. Perhaps a good way to think about this is: what role does matrices fill in math that nothing else does? What differentiates it? What is its purpose? Why would someone invent matrices?

1263:(1850) J. J. Sylvester in London, Edinb. & Dublin Philos. Mag. 37 369 "We‥commence‥with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order." 3258:

comprehension should be able to distill the following from this sentence: Matrices are used in a field called "linearv algebra" to represent some kind of relation called a "linear transformation" between objects called vectors. To help the lay reader it may be an idea to give an explicit example of how a matrix can represent a 90 degree rotation of the plane. Although we should also avoid the pitfall of trying to explain everything in the lede.

31: 3584:

I don't have a strong opinion on the picture. I disagree, though, with putting these details about the rotation matrix. First of all, this is practically impossible to understand for a newbie, and also is unnecessarily detailed. I could imagine having a picture that shows the linear transformation (but I would use a matrix all of whose entries have different absolute value) attached to a 2x2-matrix could help in the lead.

4873:. As indicated by the sources supplied, it relates to the connection between function spaces and matrices, and as a particular example, to the connection this provides between numerical analysis, operators upon functions, and matrices. Of course, there is a connection to infinite matrices in the case of an infinite dimensional set of basis functions, but the subject here is larger and not coextensive. 5423:

finite matrices are perfectly acceptable in some applications without introducing approximation. And again, this remark is irrelevant to the discussion of the proposed contribution. And finally, your peremptory dismissal of my contributions and forecast of their ill effect now and in the future are uncalled for, uncivil, and a breach of the decorum needed for cooperation in building the encyclopedia.

5912:, we have just said that it would be better introduced in other sections. You keep asking "what is wrong/confusing with what I wrote" and I don't see any real acknowledgement on your part of our answers to your question. The answers were (again): the suggested passage borders on straying too far from the title topic; some quirky phrases (which are probably a product of your background, but this 1497:(ec) I don't think you will be insulted for that. That's the smoking gun. Having seen a lot of nonsense that was passed from one author to another, sometimes over several centuries, I think I understand what Wvbailey was trying to do. We must make sure to be pretty conservative about claims that look as if they came up via a telephone game ("one could speculate that" -: --> 6219:

articles, and since it is correct, it would not generally make sense to change it in those articles. Usage within an article should be consistent, and it is inappropriate to make sweeping changes. My position is merely that it is inappropriate to introduce a new plural (plus inconsistent usage, as it happens), and was thus reverting it to its earlier consistent usage. —

1705:. The very first sentence must simply say what a matrix is, with the only purpose to provide a method to tell a matrix from everything else. And the article does it. The only problem is that a matrix (in mathematics) is not exactly an array of numbers, but an array of "entries" or "mathematical objects" which are typically, but not always, numbers. 6196:, should be changed (or footnoted as Rschwieb suggests) as well. Your position forces you to scour Knowledge searching for "formulas" and replacing with "formulae". The role of Knowledge is to inform, not to conform, and allowing for all possible spellings, rare as they may be, would help us focus on contents rather than wasting time with form. 1178:] which gives alvus, gremium, uterus, venter, and volva, but not matrix. In fact matrix does not seem to be a Latin word at all; it seems more like retro-fitted to a Latin form. The French word "matrice" (in the sense of "mold") is quite ancient, goes back to at least 17th century, possibly 13th century; my guess would be this served as model. 5616:

infinite dimensional matrices. The content should use sources that talk about something relevant to the topic. Dragging in infinite dimensions is not relevant at the level of this topic except in a special section about that. The applications section in particular should deal with finite matrices and very clearly show how they are used.

3111:

choice of a basis, for both the domain and the codomain, just as an array of numbers only represents a vector with respect to some choice of a basis. But we cannot get into all this in the lede -- already we hear objections that a) the lede is hard for non-mathematicians to read and b) the lede does not say what matrices are good for.

2559:

looked in about 20 other books on applied math, theoretical physics, operators, elementary abstract algebra and suchlike -- typical of someone trained in natural sciences 60+ years ago who has used special functions and linear algebra since then. They use element exclusively. Will post this and continue with some concerns, unindented.

1526:(the Merriam-Webster reference says "female animal used for breeding, parent plant"). And I still find it curious that in Latin one would use a female suffix -ix with the stem matr- that is not lacking in female connotation (if ever I need to coin a term for a new mathematical notion, I will certainly consider "patrix"). But bravo to 5726:

section. Mathematically trained editors might have found some phrases to be (mathematically) nonstandard (such as "superposition", which appears to mean "linear combination"). Also mentioned before: the list of five questions is unencyclopedic. There appear to be sufficient references to numerical methods without the one you added.

3333:", because not everybody will understand it. But this is a serious encyclopedia, not a friendly textbook written for high-school students. I think this sentence gets to the core of what matrices are, in a concise and (compared to most mathematics articles) clear way, and so it should be one of the first sentences of the article. 2428:

than just a book full of words. And the opening sentence doesn't indicate that in mathematics the term "matrix" has more meaning than simply being an array of things.. I think this is the problem the anon was having. Notice that both the German and French Wikipedias mention this extra structure in their opening sentences.

1131: 6267:

not basing anything on anecdotes: I am quoting from widely used dictionaries such as OED and Webster. On the other hand, by running stats on google, you are conducting lexical research, which is another thing I understand we are not supposed to be doing here on Knowledge. (Note that you misspelled "occurences" ;-)

4360: 1326:

determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order. [If you have it, and only if you have it, can you could add e.g.

1128: 6005:. First this is not an advanced mathematics article but a very elementary one, so definitions should not depend on an advanced understanding of mathematics or use of symbols. Second your definition was wrong: matrices are not restricted to just real and complex numbers but can contain any sort of thing. E.g. 4207: 3141:

polynomial functions and their compositions, which (1) as such do not play a very central role in algebra, and (2) the role they do play is in no way similar to that of matrices. Saying y = 2x is a linear function acting on real numbers shows what you meant, but is a very sloppy use of language. I'd say

6230:

I am aware that "matrixes" is much less frequent than "matrices" and did say that in the part that you obliterated (without asking). But, whether you like it or not, and small as it may be, the percentage is not 0. I thought it is Knowledge's policy it is better to state a fact than not, especially

5417:

Timothy: It is unfortunate that you have decided further discussion will not be productive. I am unsure what the basis of that decision may be. In case some response to your remarks might be of interest to others, I'd say the following. Your sharp remark that "function spaces ARE function spaces" and

4809:

Small miff in the article, why does the polynomial change from λ to t? Would we be better off editing "The function pA(t) = det(A−tI) is called..." into "The function pA(λ) = det(A−λI) is called..." or is there a common use I'm missing? My relationship with eigenvalues has been long-distance at best,

3653:

Anyone who comes to this article trying to learn what a matrix is used for will not understand "linear transformation of vectors in linear algebra". This explanation fails to meet the requirement of accessability. That Knowledge is "not a friendly textbook written for high-school students" does not

3257:

This has the advantages of saying exactly what matrices are used for. Of course, we may have the issue that a lay reader has no idea what a vector or a linear transformation is. But even that reader will get more information from this sentence, then the current enigma. (Any person with normal reading

3060:

I've done some work on the lede, fixed a few typos. The strongest impression I got, while editing the lede, was that it gets too technical too quickly. I think a lot of it can be cut. Advanced topics are better covered in the body of the article, or in articles of their own. But I did not want to

2855:

I disagree with merging that here. The article on matrices needs to focus on what is most important. Matrices with one entry are hardly ever important, which is why the article cannot tell much about them. If we start adding every possible definition related to matrices in this article we end up with

2616:

Only trying to point out that coverage is very thin and non-representative. Needs informed thought and planning to build on these comments. Many people would consider this higher priority than working in high order arrays. Hope this triggers more input from people who have taught and used matrices --

2573:

I'm not sure about this.. to a mathematician, a matrix itself is often considered as an "element". If you're referring to the individual entries in a matrix, then the word "entry" seems way less ambiguous.. because "a matrix element" could either refer to the matrix as a whole, or some specific entry

1794:

Well, further down, the article does have a section "Matrices with more general entries". Getting into too much detail about this at the start of the article seems like a bad idea from the perspective of accessibility. And we better write this article in such a way that an average middle schooler can

1681:

There is room for improvement in the rest of the opening paragraph and lead though. I'm not sure about having the example of a matrix in the second line. It breaks the flow of the first paragraph making it harder to read. Would it be possible to somehow move it to an infobox? Similarly, I don't think

1675:

I don't think I agree with the assertion that the current first line is particularly boring. It succinctly and correctly tells the reader exactly what a matrix is. (I think the IP who started this thread is partly victim of the confusion of thinking that a matrix is more that just a rectangular array

1563:

This is all very concrete, and I'm sure it's useful, but I don't think it's conveying what matrices actually are. What kind of thing are they? Are they some kind of "construct"? Are they like other things? Challenge: explain matrices without describing what they look like or how they're used. What is

1446:

You're right, this is a tiresome discussion: you've made no case whatever for your OR extrapolation of the usage of "matrix" to "generating from a womb" w.r.t. application to the above quote from Sylvester, where neither word "generating" nor "womb" appear, and you clearly are not open to the notions

266:

In all and every (printed) book or article on matrices in English I've seen, the thingos at given positions in a matrix are called "entries". The same is true for this article, but without defining "entry" explicitly. I've heard rumours of the usage of "elements" instead; and I notice that this usage

5422:

of coefficients and makes no statement whatsoever about functions and their relation to vector spaces, which is not pertinent to the description of how correspondence of operators to matrices arise. Your later observation regarding finite matrices as cut-offs of infinite ones simply is incorrect, as

5353:

Operations in function space can be mapped into operations in a vector space using a basis set of functions. This point is not made elsewhere in the article. It need not be an infinite basis as you all seem to think; for example, in group theoretical treatments of atomic and molecular spectra finite

3703:

Although the opening paragraph should address a broad audience, I don't think this is achieved by trying to explain terms like "linear transformation" that are (one hopes better) described in their own article. If the essence of what matrices are useful for can be explained without mentioning linear

3599:

I find the current version acceptable, and agree that the version in a box above is too complicated. There is still a small technical problem: y = 2x + 3 is a linear function that does not generalize to a linear transformation, but I can live with that. It would be ideal to have a brief animation

3583:

I don't mind moving the linear transformation bit up, but it needs to be in line with the rest of the lead. Moving just one sentence as someone did earlier simply creates a disconnected text. Also, sums of matrices are easier to understand than products (and are also treated earlier in the article).

3140:

OK, thanks for the reply which clarifies a lot (and which I'm reading somewhat late). The confusion was not so much about the word "algebra", in so far as high school algebra is actually a part of algebra, but about the word "function". I was thinking of such functions as exponential, trigonometric,

3032:

linear functions. In fact, this use is so common that people often misuse the terminology and not make the distinction between a linear map and the matrix that represents it in a certain basis. This is a form of jargon, which should be avoided as standard use in an encyclopedia, although this jargon

2520:

Many thanks for your comment. It encourages me to go on for a bit, albeit restricted to books I have at hand -- cannot get to library for several days. As regards "element" or "entry", I have now found in my copy of the MIT Press 1993 edition of Ito's 2-volume Encylopedia of Mathematics, section 269

2427:

a matrix"! Can I ask why nobody responded with: "You cannot be told what a matrix is.. you have to see it for yourself." But seriously, the anon has a point.. after all, a matrix in the sense of this article is more than just an array of numbers or symbols.. in the same way that a dictionary is more

1645:

a matrix! Matrices can be (are) used to represent linear maps, but are not themselves linear maps. A matrix is nothing more or less them a rectangular array of entries (most of the time numbers). In practical classroom treatments they two are so intimately related that the words "matrix" and "linear

1248:

Etymology: < classical Latin mātrīc-, mātrīx, female animal kept for breeding, in post-classical Latin also (from early 3rd cent.) womb, source, origin, apparently < māter mother (see mater n.1), with alteration of ending to -trīx-trix suffix, perhaps after nūtrīx wet-nurse, nurse (see nutrix

6218:

You seem to have missed the point. No-one has suggested any such "forcing". We are simply saying that it is inappropriate to modify the article to emphasize a word usage that is so minor; in a dictionary it would be appropriate to list all accepted plurals. The plural "matrixes" may be used in some

5985:

In the introduction an explanation in terms of rectangular arrangement is already given. This however is not a definition, but merely a description to make the notion understandable. This also holds for the so called definition in the article. The definition I gave, however, is a formal definition,

5819:

The resistance to these proposals has not taken the form of suggesting how the article might adapt to include explicit reference to these extremely important applications, but has instead taken a defensive stance that the present proposal to mention functions explicitly is not just unacceptable but

5740:

I misread what you meant by completeness. No, I was referring to the crowd of cryptic special terms: the Galerkin method, the finite element method, the so-called hat functions, the triangular functions, Hilbert space, Banach space, generalized Fourier series. While it's good to give examples, this

5457:

Certainly it's impossible for me to help you improve what you've written: to do so I would have to understand it and I can't (and I have a degree in maths). More generally it is not the purpose of WP to help you improve your mathematics writing. If you are not experienced enough in a topic to write

4920:

Perhaps you could suggest some revisions, or attempt writing something yourself? The objective is to provide guidance to the other articles on WP without going into massive detail or arcane jargon, as is so common in the math article of WP, and to provide some sources, as is so uncommon in the math

4625:

This is no bug, it's a feature. And not only of texvc, but also of the original amsmath. Since the number of columns is not predefined, 10 columns are always allocated and the actual number used is determined during the interpretation. To use more produces an out of bounds error. There is a special

3122:

the study of matrices is so important, something that I didn't understand as an undergraduate (to my sorrow) and something many of my students need to have explained. Certainly, the idea expressed above by Marc van Leeuwen, that they "can be written down", is close to the mark, but conveys nothing

3110:

I undid the recent revert. Please see the discussion above. In a first course in Linear Algebra, matrices are introduced as a way to represent linear functions on a vector space. The only problem with saying just that is, of course, that they only represent a linear function with respect to some

3076:

A recent addition to the lead leaves me speechless: "One major use of matrices is in linear algebra, where they play a role that is in some ways similar to the role played by functions in algebra." My first impulse was to throw out the sentence as original research, but it might be more instructive

2886:

entry. I know that we need to respect the literature, but this terminology is none the less highly misleading: obviously, the "single-entry" matrix does not have a single entry (see example)! I disagree with merging, but I suggest to rewrite the definition: "A matrix with a single non-zero entry is

2558:

So I think my use of element safe. If the preferred term is a big deal someone might like to check Sylvester's paper considered by Morris Klein to be the first use of the term -- in case the article does not give it, the reference is Phil Mag (3) 37 1850 363-70 (Coll math papers 1, 145-151). I have

2487:

I was led to this article today by accident, and was startled by the presence of the term "matrix entry", then saw this had been questioned, and thought the questioner entitled to informed follow up. Hence the section preceding this. When I started to accommodate the preferred usage of "element", I

2229:

Scalar can mean a number of things. One of them is a rank 0 tensor, which can be represented as a 1x1 matrix. In this case, one might even say that the scalar is the 1x1 matrix, since this representation does not depend on any choice of basis. Scalar multiplication (of matrix), can be thought of as

1779:

of the article now plainly contradicts the lead by saying that the entries of a matrix must be numbers. Of course one can find any number of textbook authors that give simplified statements like this for the sake of pedagogy. But it's not good. What to do about it? (Replacing the definition and the

1286:

Your quote from the OED supports the first half elegantly. But any inference to "wombs" is not supported by the quote, nor its context. Here's why: The word, at least in my Websters has the usage 2a. 2b. the natural material in which a fossil is embedded, material in which something is enclosed. I

6266:

Thank you for clarifying (it was not needed though, I did note your numbers) but as with all stats you need to agree on a threshold to declare something insignificant. If there is such a threshold, please quote, otherwise your unilateral decision amounts to a point of view. Please note that I am

6148:

Since the blasted editor submitted changes when I pressed the right arrow key (WHY??!!), I was unable to finish my edit summary. I was about to say that googlescholar had the ratio of occurences of "matrixes" to "matrices" at 103k to 2330k. That is, of all those hits "matrixes" accounted for about

3012:

I agree with the comment in the "This is Great" subsection. While there are examples of matrices that are simply arrays, the Alexander matrix of a knot for example, in most cases matrices are linear functions, and the lede should say that. The question is, how to say it in a way that is clear to

2600:

Here are terms that are mentioned in the accounts of matrixes in books that are classified as mathematics (albeit elementary), physics, chemistry and engineering: relationship to determinants (the WK article on these is bizarre), addition, subtraction, scalar multiplication, matrix multiplication,

2403:

scalars to be one by one matrices, how to you finish the sentence, "One by one matrices with entries from ...". You certainly don't want to say, "One by one matrices with entries that are one by one matrices." Since you can't avoid mentioning scalars, why not do what all the Linear Algebra books

1605:

If I knew that, I'd have added it myself. :) Everything you listed makes it sound like its inventor just pulled it out of his butt. "Well, I wanted something to multiply these groupings of numbers together in this arbitrary fashion that I just made up for no reason. Hey! Look! I can also do linear

1586:

So, what kind of thing do you think matrices are ? We can describe how they are usually written down (as a rectangular array of numbers), how they behave (rules for addition, subtraction, multiplication by a number, multiplication by another matrix), and what they are used for (representing linear

6167:

Google books puts the ratio even lower (at 2% = 194000/9740000). I would go further and suggest that this sort of language variant does not warrant any mention whatsoever unless it serves to clarify a possible confusion of interpretation. Such language variations fall entirely into the scope of a

5357:

That point established for the reader, I imagined that a reader needed to be cautioned that there were restrictions upon this process of converting operators on function to matrices. I did not feel it was the place to delve into these issues, so I posed them as a list of some questions the theory

4374:

Failed to parse (unknown function "\setcounter"): {\displaystyle \setcounter{MaxMatrixCols}{14} \begin{bmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\a & b & c & d & e & f & g & h & i & j

3669:

I also consider TR's suggested "A major application of matrices is to represent linear transformations of vectors in linear algebra." to much in-your-face, but the current "A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as

3336:

Currently I read the first few sentences as: "Matrices are just rectangles filled with things; look here's one (example).. see they're not too scary!" Frankly, the opening paragraph is missing the point. I think TR's suggested sentence is far superior. We should not be dumbing the first paragraph

5879:

Well my resistance has been because the addition seemed too far from the main thrust of the article. Things like Galerkin methods might be worth sticking into a see also or stuck into the note about finite element methods but this article should not have loads of stuff about things which are not

5839:

With respect that is mischaracterising and ignoring my objections. But whatever the reasons there are four editors now that have objected to your additions on multiple grounds which you have largely not addressed. You might take this opportunity to recognise that the consensus is against you and

5615:

I don't know what you mean by 'proposal' here. The text refers to generalized Fourier series, Hilbert spaces and Banach spaces. They are all typically infinite dimensional. Then it jumps to numerical approximation with no clear link. It is messy with no clear content and seems to be mainly about

1325:

Per the OED the first usage of the word "matrix" with respect to mathematics appears in (1850) J. J. Sylvester in London, Edinb. & Dublin Philos. Mag. 37 369. "We ‥commence‥ with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a

1213:

but what was it? In my working on wikipedia articles I've gone through a number of these and am involved in one now, i.e. researching back to the first usage (if possible). Almost always the results are surprising and troubling. Almost always these involve a trip to an academic library and hours

5474:

John, clearly the objective is not my improvement, but a collaborative effort among us all to make an encyclopedia. It is indeed unfortunate that you do not understand this subject, or at least not sufficiently to understand this very simple presentation, nor to write "clearly and well" on this

3092:

I'm at a loss to understand what you are getting at. "algebra deals very little with functions"??? I was, of course, using algebra in the sense a layperson would understand the word: the subject taught in high school in which most people are first introduced to functions, but even in abstract

1623:

Suggestion: "A matrix, in mathematics, is a mapping that acts on vectors, which may change both their length and direction. It is represented by a rectangular array of numbers, and acts on vectors by means of an operation known as matrix multiplication. Matrices are fundamental in many areas,

5725:

As commented before, the removed content said little more than the last two-sentence paragraph of the infinite matrix section. There is no reason to make an entirely new section for a finite dimensional special case. Perhaps you can fashion a few gems to add onto the end of the infinite matrix

5630:

Infinite matrices certainly belong as a subsection in this article. While Brews did make a passing comment about infinitely many basis elements, that does not appear to be the main thrust of his suggestion. The main problem here is that the deleted content did not actually go beyond previously

3271:

Consider this statement: "A major applications of matrices is to represent wibblefubwilda of tribbles in jabberwocky." That's what the mathematically correct formulation is going to sound like to a lay reader. True, there are hyperlinks, but a frustrated reader is unlikely to follow them. I

1521:

Just dropping by to say I stand corrected in this issue. The evidence is quite convincing. Just one minor point to quibble: it would seem that when Sylvester used the word Matrix, it was as least as common in English as in Latin (in any case he did not seem to feel the need to point to a Latin

1347:

This is not OR, no more OR than quoting any other source. But without support the fantastical synthesis of "matrix" to something "issuing from a womb" certainly isn't in the quote above, at least as you've presented it. Maybe it is there in the original, but if so we need the whole quote. Bill

5361:

Finally, the important application of matrices to numerical analysis is brought up. That is most commonly framed by using a particular basis set that consists of function localized to a small region. This approach generalizes the older one based upon finite difference approximations. It is an

4899:

that it does not belong. It is just an extension to an existing section rather than some new application, so should be linked to it and not separate. But mostly it is so poorly written it is impossible to understand. It barely mentions matrices, consists half of unencyclopaedic questions and

2447:

A SciFindScholar search for papers containing "matrix element" and "matrix entry" found 7913 and 8 papers, respectively. This is consistent with "entry" seeming bizarre in a natural science context. A Web of Science search reported 27,000 articles containing "matrix" & element", and 1800

5398:

Your point 1 is already very confused. Functions spaces ARE vectors spaces (something which isn't that relevant to this article) therefore operators on functions spaces are operators on vector spaces and can be represented as (infinite) matrices. That sometimes a cutoff can be introduced to

4064: 5383:

Of course, every reader will have their opinion of how well the ideas have been expressed. It would be helpful if the "confused" ideas were identified. My suspicion, and only that, is that the brief outline of how matrices are related to operations on functions was simply not couched in a

2464: 6359:

Asking about what threshold of use is a better question. Five percent already seems like a very generous threshold. "Not zero" is a patently bad (and absurd: there are no zero usage terms in writing) I wonder if you meant something else. Perhaps you meant that you think 1% or 2% is

5856:

John, I have made a detailed attempt above to elicit just what it is you find confusing about the proposed text, and you have not replied. I do see the consensus, although it appears to me to be driven by a certain aesthetic and not by reaction to the missing content of the article

6251:

That should serve as a second proof that relying on your anecdotal feelings has lead you to the wrong conclusions. And I am sorry if I sound gruff, I am just trying to say it plainly. I do hope you adopt some similar reasonable measure before you make edits of this nature, though.

5924:

To make my earlier suggestion more plain, I think it would be a good idea to take the tangle of applications you mentioned in the suggested paragraph, locate places where they fit in, and use the example to spruce up that place. We could tackle these new additions one at a time.

3931: 1428:

That's because "womb" was a popular meaning for "matrix" at the time. And a number of RS linked above (see my first reply) should suffice, as not a single RS seem to be contradicting this. I don't see any other logical reason to continue with this historical OR on this pages.

1659:

What you say is true, of course, but I'm trying to come up with something that doesn't put readers to sleep. How about, "In mathematics, a matrix is one way to represent a mapping that acts on vectors that makes complex calculations simpler." If not, please suggest something.

1175:

Wvbailey would do better to use polite language. But as for the matter of Latin, I must agree that there seems to be no justification to the claim that "matrix" is a Latin word for "womb", however many math books say so (probably the authors are copying one another here). See

5884:

guideline ssays "To demonstrate that you are not adding OR, you must be able to cite reliable, published sources that are directly related to the topic of the article, and directly support the material being presented (and as presented)." anything much beyond that goes into

4939:

The section was very badly written, and consisted mostly of very confused statements. The only relevant part to this article is that infinite matrices can be used to represent linear operators on infinite dimensional vector spaces (such as function spaces). This is already

632: 5362:

immediate outgrowth of the earlier discussion of how a basis leads to matrix representation of operators on functions, in particular differential operations. There is nothing about numerical analysis in the article on matrices, although it is a very major use of matrices.

2785:

It doesn't seem notable enough for its own article, and it's not an important enough topic to be discussed in a main article about the subject. The main article should be even more selective than our general criteria, since there are potentially thousands of topics that

2366:

I think that the scalars can be naturally identified with the 1×1 matrices. One could say something along the lines that though scalars are not technically 1×1 matrices, the two concepts are often conflated since confusion is unlikely to arise (or something of the sort).

5787:. However, there seems to be no reason to refuse to spell out the particular application of matrices to function spaces by noticing (i) that a set of basis functions serves to map a function into a column vector, (ii) a set of basis functions serves to map an operator 1307:

in together with the supposition, but only as evidence, not raw, brute fact. (I'd say the only admissible evidence is what Sylvester actually wrote. If he wrote "I think of Matrix as a womb from which the determinant issues" then I'd say you could report that as brute

925: 1463: 5939:

I have no further interest in this article. I have made an effort to point out its shortcomings and written a subsection to meet them. No attempt has been made to suggest a means to address these problems, and only argument is forthcoming. It's a wasted effort.

5782:

on the grounds that this material duplicates what already is present in the article. In an abstract sense that is true for a set of functions satisfying conditions like (f+g)(x) = f(x)+g(x) and (cf)(x)=f(cx) is a vector space, which is the focus of the article

5157: 4879:

This function-space section is short, and of course much more could be said. If the links to other WP articles and the sources provided are not felt to be sufficient guidance to these topics in this overview, then the section should be enlarged, not deleted.

2957:. Some good arguments were made, and I see consensus will not be possible anyway, so rather than continuing the argument I'm going to withdraw my proposal. If someone else feels continuing this discussion might go anywhere feel free to open the discussion. -- 3287:

Lay readers are not helped at all by being vague. Moreover, referring to what students learn in XX school is not helpful to the general audience, since there is a large part that both has no clue what matrices are and has no clue what is taught when in what

3123:

to the layperson, who would naturally ask, "Who cares? Lots of stuff 'can be written down'." The best I've been able to come up with is to make an analogy with functions, something most readers have at least heard of and may understand the importance of.

3684:

P.S. To be honest, I don't think these elongated discussions about the lead of this article help the article improve much. It is certainly not a perfect article or lead section, but at least a GA. There is much worse than this we should focus on, I think.

5880:

related to the topic. I think this has been confusing 'uses matrices' and 'is about matrices' Galerkin methods are not about matrices. Any mention would have to be in the application section and we should not go into much detail about the innards. As the

4868:

using the one-line comment: "remove rather confused section on a subject that is already covered in infinite matrices section above". Aside from the fact that there is nothing confused or confusing about this section, it certainly is not covered by the

2160:. (Indeed MATLAB uses a less strict definition of the matrix multiplication, represented by a function called MTIMES, which performs scalar multiplications when one of its operands is a scalar, but I believe this is not 100% correct in mathematics) 2019:

I do not want to impose my (longer) version. I highly value the opinion of other editors, especially if they are so talented as those who wrote this article. I just want to point out that some readers might not understand the current version.

4219: 4626:

macro in amsmath to increase the number of preallocated columns. But due to the wiki-structure, that probably won't work here. A way around is the blockwise construction of large matrices. Or the use of the old-fashioned array environment.

1447:

of evidence and reason. The only achievement here has been for you to actually do some decent research to validate the date (it's changed how many times?) and provide us with a decent source and a nice quote. My role here is complete. Bill

5916:

a math article after all); the list of questions is unencyclopedic and can't be included; the added material is only superficially "new" and doesn't deserve its own section. You acknowledged the last point, but mainly with a rant about

5956:

I think it would be a good idea to take the tangle of applications you mentioned in the suggested paragraph, locate places where they fit in, and use the example to spruce up that place. We could tackle these new additions one at a

4900:

sentences of links, and even where it tries to explain it is so badly written it's difficult what it's trying to get at. This is a good article and it falls far below the standard of writing expected for such, so does not belong.--

1197:. Actually when I saw the original entry with the bad date I was going to delete it as BS dropped on the page by a vandal, but as it was entered by a registered wikipedian I gave it the benefit of my extreme doubt and resorted to 4076: 1676:

of numbers.) Of course, it is not the most exciting phrase, but not particularly boring. Any delay in telling the reader what a matrix is, will only serve to confuse lay readers. As such I don't think the first line needs fixing.

1137: 5418:

using that remark as evidence that the proposed contribution is inaccurate is first of all unkind as the remark you address is not part of the proposed contribution. The proposal points out how a function can be mapped into a

6187:

What's wrong with "minimums" as an alternative to "minima"? It's perfectly legitimate and should be included. Is there a Knowledge rule that says alternative spellings should be censored out? I don't think so, otherwise

3808: 3427: 2636:

covered in the article. About the choice of applications: there are tons of applications and in the GA process we chose to present a little bit in depth a few select applications instead of just mentioning whatever is out

1366:

I am getting tired of this debate, and especially of occasionally rude language addressed at me. Somebody else should join. Your selective Websters are not doing a good job. Here is the first definition for "matrix" in

5586:: Agree with the removal of any expanded details about infinite matrices in this article. That's a different subject under Hilbert spaces and suchlike. This article should confine itself in the main with finite matrices. 2151:

environment (as you probably know, MATLAB means MATrix LABoratory, not MATh LABoratory), scalars are defined as 1x1 matrices, but I think this is not 100% correct in mathematics, because in mathematics the definition of

1940:

I agree with Marc van Leeuwen that simlifying the intro is desirable, but possibly one of the paragraphs was over-simplifyed. In this paragraph, the expression "Arrays of values of more than two dimensions" is unclear:

407:

Secondly, it isn't possible to multiply Ax if the x's are row matrices, since the number of rows in A (i.e. 2) does not equal the number of columns in x (i.e. 1). As written, Ax is not a valid operation, only xA is.

6097:

I don't see any major deficiency in the description as it is now. There is no reason to go out of the way to make it more technical. It goes without saying but: WP is not always bound by the same rigors as a textbook.

5505:

To repeat, perhaps you can bring forward some specific matters that you did not understand. For example, do you find confusing the point that subject to some requirements examined later, a function can be expressed as

403:

First of all, at different places in the section, different sets of vertices are given for the parallelogram, some with (a,b) and some with (a,c) as a vertex. The picture suggests that (a,b) should be the right one.

3943: 1901:

Thank you for pointing this out. I agree with you. I don't really care to explain what arrays are in computer science. I only want to be sure that we don't say something wrong, i.e. that multi-dimensional arrays are

812: 2750:

I don't find the content to be worth merging (or even really keeping). "Single-entry matrix" doesn't seem to be a term in wide use, and it's clearly not important enough to be treated in the main matrix article.

2505:

Thanks for the very welcome information. I think scientists are more apt to use "matrix" in the geological sense, and not be bothered with "element" in the set theory sense. Mathematicians, the other way around.

2398:

Yes, if we have a field of scalars, then we can use that to define a field of one by one matrices with entries from that field of scalars. But if you try to avoid mentioning scalars as distinct from matrices, and

1381:

The womb; the uterus of a mammal. Also (in later use esp. of an oviparous vertebrate or invertebrate animal): the ovary; the part of the female reproductive tract producing or storing eggs or embryos. Now chiefly

2104:

I believe that the third option should be avoided. In general, too technical terms or expressions should be either avoided or explained, in an introduction. About this, I invite everybody to share their opinion.

2070:

I agree that is better not to mention "arrays with more than 2 dimensions", than mentioning them without explaning what a dimension is, but there's also another option: explaining it (e.g., as I proposed above).

483: 294:. Actually, at a glance, I found no direct mention of "entries" anywhere in that article. However, there was an indirect reference, which actually might be a bit confusing for non-expert readers. In the section 5889:

territory. If I was pedantic I would ask for a citation saying something like 'Galerkin methods are an application of matrices' but I'm not asking for that. I just want the article to stay reasonably on topic.

3553: 1882:

are nowhere mentioned in the article (rightfully, I think), hence the lead should not do so either. If you disagree with this non-appearance, I suggest first working on the article, then on the lead section.

3820: 2771:

I'm not sure what you are saying. If it is important it should have an article. If it is not, it should not have an article. Are you saying that it is not important, so it should not have an article? --

1818:

I've been cleaning up the beginning to make it readable. I moved the plural parenthetical explanation to the next paragraph. I think it's definitely an improvement, but Jakob.scholbach wants to revert.

5343:: "The only relevant part to this article is that infinite matrices can be used to represent linear operators on infinite dimensional vector spaces (such as function spaces). This is already covered." 673: 542: 2316:

I agree as well, to avoid confusion we should try to use the word scalar only in one sense of the term. I do think that the article should mention the Kronecker product somewhere (not in the lead).

121:

You can define them, for example, as a collection of vectors, or as a description of a transformation. But you're right, the current introduction is the best for people unfamiliar with matrices. --

5601:

Dmcq: Exactly where are you coming down here? The proposal refers to finite matrices, but perhaps you are not aware of that? I don't know if you support the proposal deliberately or by accident.

3570:

The current picture would be moved to the notation section. The use of row vectors rather than the more common column vectors is not ideal, but does make it easier to mention the vectors inline.

968: 817: 411:

It would also help to clarify things if the transpose operations were kept separate from the multiplication operations. It makes the article unnecessarily confusing to have them interspersed.

6113:

Who says it should? But the so called definition is no more than a repetition of the description in the introduction. And in an encyclopaedias also a formal definition should have its place.

2090:

The sentence about "Higer-dimensional arrays" has been reinserted (at the end of the introduction). We already discussed about this, and decided to remove it. There are three possibilities:

5816:

There also is no reason to avoid pointing out that the numerical solution of differential equations can be accomplished using matrices arrived at by the use of localized basis functions.

6021:

in the first row, real numbers in the other two rows. Third, ignoring the difficulty and correctness of it, your version had some basic grammatical errors that made it incomprehensible.--

1029:

That's right. Not only did this section contain some inaccuracies, it also did not fit well into the article's section ordering. I merged (and corrected) its content with other sections.

231: 1160:

Gimme a break. As typed into the article this was utter crap and deserved to be reverted. Whether or not, after the date correction, it's still true is a matter for the historians. Bill

5053: 1270:

I say there is a clear need here acquiesce to authority, rather than try OR on this. I will restore the original quote as without it being present on the page the discussion is moot.

6301: 1049:

I have added parentheses to the example calculation to what I think is clearer, since I had to spend some time trying to disentangle the dots and the plusses. I also think that (now):

3360:

I do think it is a good idea to have the prime use of matrices, representing linear transformations, as early as possible. How about the following suggestion for the first paragraph?

317:

usage; not even the accountant's entries are mentioned. (Actually, I think that the accountant terminology is the historical reason for the term entry in connection with matrices.)

162:

This seems to be a good point, but maybe too fine a point for the lede. In knot theory, the matrices I work with have polynomial entries. That should be in the article somewhere.

5402:

Your point 2 demonstrates that you still have a limited grasp on encyclopedic writing. Which is to bad, because it means that you are likely to continue doing more harm than good.

740: 6298: 6291: 103:

I don't think this is a good explanation for somebody who does not yet know matrices. If you define numbers via matrices, how do you define matrices then (without using numbers)?

84:

Does anyone have a good idea what to write in the introduction? In higher level mathematics, and mostly in system theory, one of the most important things is that a matrix should

6304: 6295: 5559:

Or, perhaps rather than have the mathematical requirements pointed out as questions for the reader to pursue in other WP articles, you would like more detail in this paragraph?

5358:

provides, and provided links to WP articles where answers and further discussion could be found. That seemed appropriate in this article, which is not about function spaces.

4355:{\displaystyle {\begin{bmatrix}1&2&3&4&5&6&7&8&9&10&11\\a&b&c&d&e&f&g&h&i&j&k\end{bmatrix}}.} 320:

What I wonder is, first, is actually "elements" nowadays used in lieu of "entries" in some text books or articles in English, at lower or higher level; and second, is there

1005: 6356:"matrixes") so I hope we can skip it in the future. Mocking an obvious litmus test for commonality of a word is also a waste of time that does not advance your argument. 5489:

I understand matrices very well, and use them in my work on a regular basis, and understand the article as it now is. But I could not understand what you had written.--

4572: 1007:, getting the same set of vertices that xA returned, but represented as columns instead of as rows, which can be easily transposed back into the row matrices equal to 4956:

It would be helpful if you all could provide specifics, rather than generalities. For reference the section is reproduced below. I'll make some suggestions under it.

4781: 3096:

In any case, I hope the explanation that "algebra" means the subject taught in high school, not abstract algebra, explains the purpose of the sentence you object to.

5653:: Agree with removal of section "Functions and function spaces". It did not add anything new, it seems mainly to be a lot of name-dropping. It was also quite vague. 3237:

The last sentence of the first paragraph is currently extremely vague. I fear it may be mystifying to lay and expert readers alike. What wrong with just saying:

4202:{\displaystyle {\begin{bmatrix}1&2&3&4&5&6&7&8&9&10\\a&b&c&d&e&f&g&h&i&j\end{bmatrix}}.} 5372:

I have set out my objectives here, which seem to support the claim that this work is not simply an unnecessary elaboration of the section on infinite matrices.

5346:

Although this seems to be how this section comes across, that is not an accurate portrayal of what I wished to say here. There are several points being made:

5631:

presented material. It was removed because it was superfluous. There may be a nugget or two which would go well at the end of the infinite matrix section.

2254:

Interesting, but then again, a Kronecker product is not a matrix product. I think it is safer not to mention scalars or 1x1 matrices in the introduction.

1256:

It also lists couple of dozen examples. Here is the oldest quote OED is using in the "mathematics" section, which actually supports the historical claim:

4642: 2891:

called a single-entry matrix, and denoted as follows:...". However, we might need a section about "special matrices", (with reference to main article:

353:

for matrices as tensors, I found that here the entries indeed are called "elements". Now, it's about 40 years since I had Shilov as a text book; but I

147:", although I'm not sure that expressions is the right term - the elements of a matrix can (and often are) anything from numbers to other matrices. -- 5177:

What limitations are placed on operators upon these functions to insure they result in new functions that also are expressible using the basis set {φ

5458:

clearly and well on it then you should not be editing that topic; there are many other topics, and no shortage of articles in need of improvement.--

1834:

Sorry, I don't agree with you. I prefer the previous version, now restored by TR. It is more complete and has a more standard format and structure.

1501:"it is well known that"), and this claim did look a bit fishy. But it appears to have been correct. This source is a great addition to the article. 4059:{\displaystyle {\begin{bmatrix}1&2&3&4&5&6&7&8&9\\a&b&c&d&e&f&g&h&i\end{bmatrix}}.} 3747: 1795:

understand at least the first few sections of it. May be some sort of note on the first definition that points to the section with generalizations?

3369: 2171:, as far as I know, does not meet the (strict) definition of a matrix multiplication, although it can be represented by the matrix multiplication 2610:

From books on numerical analysis -- inversion and diagonalization -- major approaches, impact of high precision arithmetic. Strassen algorithms.

1624:

because they provide a relatively simple and broadly aplicable way to deal with complicated problems in mathematics, science, and engineering."

1295:

that the word "substrate" is the intended meaning (synonym), not "issuing from a womb". But this is just my supposition, I cannot support it by

2613:

Field of application -- present mention of geometrical optics very selective choice. Quantum theory host of applications. Likewise statistics.

1646:

map" are (almost) used as synonyms, but that does not make them the same thing. In particular a matrix is not "a mapping that acts on vectors".

1483:

That is a very clear and unambiguous argument in support of all the sources I listed above. I will try to fix the statement to address this.

5369:

The above discussion is supplemented with two really excellent books that cover much more than the points they are cited in connection with.

5162:

Needless to say, there are a number of mathematical requirements to make sure this idea works. For example, these questions are addressed:

6028: 5847: 5496: 5465: 4907: 749: 1828: 1669: 1635: 1579: 2918:

The "single-entry" matrices are, in my opinion, not important enough to be listed in this section. They should, however, to be listed in

6069:

Since this encyclopedia is written in English, rather than mathematics, that limitation of the language of mathematics is not a problem.

4811: 3342: 2994: 2575: 2429: 71: 59: 5695:

was something not found in the article. Perhaps you see connections I don't see with the rest of the article. Could you point them out?

3926:{\displaystyle {\begin{bmatrix}1&2&3&4&5&6&7&8\\a&b&c&d&e&f&g&h\end{bmatrix}}.} 427: 4613: 3722:

I already said above that the sentence "...linear transformations, that is..." looks good to me. I agree that is is time to move on.

1607: 1571: 1014: 122: 89: 6168:

dictionary, which Knowledge is not. Otherwise, for example, we'd have to list "minimums" as an alternative to "minima" as plural of "

3499: 1303:. If "hard evidence" is available, e.g. one of your sources above states that this womb-business is the case, then you can put that 5437:

Timothy, your remarks are not directed at improving the proposed contribution, and do nothing to discredit it in its present form.

5405:

I have not interest in discussing this with you any further. (As the past has proven that those discussions are seldom productive.)

2404:

I've taught out of do, and make a distinction between scalars, which are the entries in the matrices, and the matrices themselves.

5723:

It's unreasonable to try to be complete within a humble subsection. That section was destined to be no longer than it already was.

3600:

that illustrated the action of a matrix on a geometric object such as a cube, but that is beyond my ability to program? Anybody?

627:{\displaystyle {\begin{bmatrix}0&0\end{bmatrix}},{\begin{bmatrix}1&0\end{bmatrix}},{\begin{bmatrix}1&1\end{bmatrix}}} 2463:

Quite generally, it is a bad idea to count hits somewhere online. Instead, one must consult authoritative sources, such as, say

5918: 5672:

offers nothing that is not already in the article. I thought the reference to numerical analysis and to matrices as related to

5475:

topic. However, you could bring some perspective forward as to the confusion this article raises in your mind as a lay reader.

1287:

know in engineering a "matrix" can be used to describe a substrate, e.g. a woven cloth of a circuit board. And considering the

1923:

Array and Tensor are not synonyms. The relation between an array and tensor is akin to that between a matrix and a linear map.

5315: 5285: 2423:

I love the original post in this thread. The anon is voicing the very natural desire to understand, by asking "Yes, but what

247:

I use that notation when I teach Linear Algebra, when I introduce matrix multiplication, but it isn't in the textbook I use.

2797: 2758: 920:{\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}a\\c\end{bmatrix}},{\begin{bmatrix}a+b\\c+d\end{bmatrix}}} 305:

Moreover, there seems to be no way for a non-expert to find a reference to matrix entries; at least, I found none. The page

5778:: Several editors (Timothy Rias, Blackburne, Rschweib, and Dmcq) have objected to the proposed inclusion of the subsection 3634:

I think most editors are aware of that, so I'm a little confused as to what point you are try to make by quoting that here.

3118:. It is a rectangular array of symbols, usually numbers. But I have not been able to come up with a way to say precisely 2136:(bold font). The problem is that, if you call it a scalar, then I guess you cannot call it also a matrix. The two concepts 2006:

one-, and two-dimensional arrays of values. Arrays with more than two dimensions (e.g. three-dimensional arrays, shaped as

1143: 5959:

Every conceivable step has been taken to accomodate you cooperatively. The offer still stands if you decide to come back.

2976:

Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

637: 2050:

I think the last line about arrays with more than two-dimensions can be safely dropped without detriment to this article.

1974:

The reader is not supposed to know the concept of "dimension of an array". I propose to modify the paragraph as follows:

6149:

4%. My speculation is also that in mathematics, the ratio is even lower. I think "ixes" is frequently even discouraged.

3219:, in other words the matrix itself (as is proper for a definition) rather than the linear transformation it represents. 2622: 2564: 2493: 2453: 1340:

states that Sylvester meant this to mean, that the determinant issues from a "womb", the Latin word for matrix. ref: -->

383: 198:

An alternate convention is to annotate matrices with their dimensions in small type underneath the symbol, for example,

930: 347: 5018:. Then an operation on the function ψ results in a new function, say χ, and therefore results in a new superposition 4841:. -- That does not say that your version is not equally correct, but it would not be an improvement of the article.-- 5821: 5152:{\displaystyle d_{m}=\Sigma _{n}M_{m,n}c_{n}\ ,\ \mathrm {or} \ \mathbf {d} ={\boldsymbol {M}}\cdot \mathbf {c} \ .} 2644:

the article. If you feel that certain topics are under-represented, rather write about them in the article (first).

2467:. He calls the things entries or components (p. 23). Your unilateral rename everywhere is inappropriate, I believe. 679:(which stores the co-ordinates of our parallelogram) in turn (i.e. xA). This gives us the row vertices , , & . 5399:

approximate an infinite matrix by a finite one isn't as much an application of matrices as it is of fourier theory.

4981: 3709: 3224: 3082: 2281: 1785: 1535: 1183: 38: 2895:) in which we shortly define some of the most important and most frequently used kinds of matrices, for instance: 6152:

Since usage is this rare, it does not seem appropriate to represent this usage in anything more than a footnote.

4825:

Hi, the polynomial does not change variables. The polynomial is defined using the (unspecific, general) variable

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What about the previous subsection? We didn't reach a consensus yet. I invite everybody to share their opinion.

5766: 2156:(scalar by vector, scalar by matrix, or scalar by N-D array) is not (always) compatible with the definition of 201: 3455:) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its 2993:

with a redirect.. and maybe with a better definition (i.e. not insist the single entry is a 1). Just sayin'..

1417:(1840) tr. G. Cuvier Animal Kingdom 40 "The foetus, immediately after conception, descends‥into the matrix." 4815: 6403:

Alan Tucker, A unified introduction to linear algebra: models, methods, and theory, Macmillan, 1988, p. 508.

6339: 6272: 6201: 3346: 2998: 2812:

From a purely practical standpoint, it is easier to merge articles that to delete one. I think the article

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we don't use, in the introduction, the expression "higher-dimensional" (or "with more than two dimensions").

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So, both "entry" and "element" do be represented in English text-books. Which term is nowadays most common?

3321:

I can understand Rick Norwood's objection to the sentence "A major application of matrices is to represent

5762: 5229: 4609: 4586: 3727: 3659: 3624: 3605: 3277: 3128: 3101: 3066: 3051: 3018: 2821: 2739: 2511: 2409: 2340: 2238:) of a matrix by a 1x1 matrix. (On the last subject, why is this operation not mentioned in this article?) 1665: 1631: 1099: 268: 252: 167: 126: 93: 6311: 6073: 5945: 5866: 5829: 5713: 5606: 5567: 5480: 5442: 5428: 5409: 5389: 5241: 5237: 4961: 4944: 4926: 4885: 4819: 3705: 3638: 3574: 3468: 3322: 3292: 3262: 3241: 3220: 3078: 3037: 2354: 2320: 2277: 2242: 2157: 2153: 2054: 1927: 1799: 1781: 1702: 1686: 1650: 1531: 1179: 686: 536: 291: 5704:? I thought discussion about that would be a digression here that duplicates material better found in 2985:

I know the above discussion was closed, and maybe this isn't the appropriate place to bring it up, but

2604:

From the Ito Encyclopedia: Kronecker products (need term direct product, too) Hamilton Cayley theorem.

1965:. Arrays of values of more than two dimensions are not called matrices, but they can be interpreted as 5524:

Or perhaps you found confusing that an operation upon the function ψ could lead to another function χ?

1201:. My Webster's Ninth Collegiate Dictionary (1990) gives the origin of matrix as you guessed it: fr L. 6352:@cerniagigante Petty sniping at spelling is not impressing anybody (especially since you are arguing 5858: 5784: 4601: 3686: 3671: 3585: 3305: 2869: 2706: 2698: 2645: 2468: 2125: 1981: 1948: 1884: 1567: 1505: 1502: 1053:(the underlined entry 1 in the product is calculated as the product (1 · 1) + (0 · 1) + (2 · 0) = 1): 1030: 182: 104: 148: 6023: 5842: 5491: 5460: 4902: 2986: 2861: 2835: 2813: 2702: 2694: 1592: 1134: 152: 6391: 1554:

If this article is meant to inform a lay person, then read on. Sorry if this sounds too critical.

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origin), and I'm not sure having seen sufficient evidence that it is used with the meaning "womb"

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Or, perhaps that when such a matrix mapping exists, it provides a representation of the operator?

5354:

sized matrices are used exclusively based upon matrices found using a limited number of orbitals.

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Couldn't agree more with TR. I reworded that bit and blended it with the remainder of the lead.

2705:

as a concept is not important enough to merit an article, and could easily be mentioned within

2002:

row or column vectors, and matrices with rectangular shape are considered to be, respectively,

1878:

It is important to keep in mind that the article's lead should match the article. For example,

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I agree completely, but how do we explain that to someone who doesn't know any mathematics?

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The literature uses the expression "single-entry matrix" to indicate a matrix with a single

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is a matrix. Of course, matrices may have only one entry, theoretically. So, a single value

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The objections JohnBlackburne raises are essentially those I had in mind when I reverted.

5380:: "The section was very badly written, and consisted mostly of very confused statements." 5233: 4846: 4794: 4633: 2632:

I don't know exactly what this post is supposed to mean. Most of the topics you point out

1879: 1862:. Also, row and column matrices need not to be the components of linear-algebraic vectors 1855: 1404:(1765) Treat. Domest. Pigeons 15 "The ovary, or upper matrix of the hen, or female bird" 5277:(Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 86 1719:

You're right and I changed the sentence to reflect that entries are not always numbers.

400:

This section has a few errors and some mixed-up ordering that make it quite confusing.

6133: 5954:

No attempt except this (twice), which I will now underline so it cannot go overlooked:

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that are superpositions of the basis set. This relation sets up a correspondence ψ → {c

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such as rotations. For example a 90 degree rotation of the plane transforms the vector

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article is one sentence, so of course, it would be merged as (at most) one sentence. --

2617:

I am sure I have omitted several topics and terms as important as those I have put in.

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We've been so focussed on the lead, that we missed that the very first sentence of the

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unnecessary, and consequently the changes it suggests are not needed. It appears that

88:

be viewed as an array of numbers, instead numbers should be viewed as 1*1 matrices. --

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consider whether further effort arguing over this might be better spent elsewhere.--

5669: 6231:

if someone bumps into that minority of articles and books where "matrixes" is used.

6128:

Have you got a citation which says something quite similar to what you are saying?

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Actually, a 1x1 matrix (with values in a field) is itself an element of a field. ;)

1145:

p. 190. Please do not rv without RS stating this is an urban legend or something.

325: 2640:

Another simple rule should be obeyed in touching up the lead section: the lead is

5824:

is present here, and that the only "good" math is pure math without application.

4876:

If anything, the section on infinite matrices is less transparent than this one.

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in 1948. He used this translation because he viewed a matrix as a generator of

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If you wish to start a new discussion or revive an old one, please do so on the

2936:

disagree with merger and agree with Paolo.dL. "single-entry" should just be in

4870: 4861: 4842: 4790: 3803:{\displaystyle {\begin{bmatrix}1&9&13\\20&55&6\end{bmatrix}}.} 1987: 1954: 4375:& k & l & m & n\end{bmatrix}. \setcounter{MaxMatrixCols}{10}} 3422:{\displaystyle {\begin{bmatrix}1&9&13\\20&55&6\end{bmatrix}}} 1127:

The sources says 1848 or 1850. It was a typo. Supported by a number of RS:

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You also refer to "name dropping", which I'd guess refers to the mention of

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I got lost in this argument. At the risk of being insulted, how about this:

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Since Matrix Theory links here, I think more theorems should be included.

5335:

Using the above, let me address your comments and request your assistance;

6054:

Frankly I don't know what a "rectangular arrangement" is in mathematics.

1215: 1194: 310: 276:

Matrix#Matrix multiplication, linear equations and linear transformations

313:, which is a disambiguation page. This page does not mention "entry" in 6193: 6169: 2716:

in April, but not followed through. It makes sense so I am redoing it.

140: 6082:

That's fine for the introduction, but not for the definition section.

807:{\displaystyle A^{T}={\begin{bmatrix}a&c\\b&d\end{bmatrix}}\,} 302:

is mentioned, but again as defined by a multiplication of "elements".

5012: 3472: 3326: 3245: 2148: 2011: 2007: 1966: 1859: 275: 6363:

Another way you could support yourself is if you found two or three

2922:. And we don't need to list all these matrices in the introduction. 2574:

in the matrix. It's probably better to use unambiguous terminology.

3061:

do too much at one time. There is still a lot of work to be done.

5271:

Gerald B Folland (2009). "§3.5 Regular Sturm-Liouville problems".

1745:

Thank you. I also appreciate the adjustments by Marc van Leeuwen.

1557:"In mathematics, a matrix...is a rectangular array of numbers..." 1091: 478:{\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}\,} 1550:

The beginning doesn't explain what matrices are (and it's boring)

3548:{\displaystyle {\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.} 1095: 6392:

http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php

2097:

we use it, but first we explain it (e.g., as I suggested above)

5195:

What are the relative merits of one basis compared to another?

4829:. λ stands for a specific value, a root of the polynomial. So 286:

as its main article; and there the same rule is given for the

282:

of a product matrix is given. The section, however, refers to

25: 5244:

to be localized to small regions, for example, the so-called

1291:

Victorian/Edwardian interest in paleontology I, Bill, would

6367:

that support the usage in a mathematical context. Regards,

6009:

of the cross product is the determinant of the matrix with

3151:

is a linear equation involving two (real valued) variables

1549: 1193:

You're correct, I should have used the more scholarly word

5562:

John, just where does there arise confusion in your mind?

6290:

In cases like this google's ngram viewer is your friend.

3183:

with rational entries can be defined as a map (function)

2533:

is a ring or a field), BUT, a little later: "The element

2335:

A scalar is an element in a field. A 1x1 matrix is not.

1994:, as they are typically used to define the components of 1961:, as they are typically used to define the components of 1084:

Maybe the date should be 1848, but even then I doubt it:

267:- inconsistently - is employed in some other articles in 80:

A better introduction than "rectangular array of numbers"

1780:

reference by Bourbaki(s) does not seem wise either ;-).

1682:

the first paragraph of the lede should discuss notation.

6189: 6002: 4984:, and their generalizations. The basic idea is that a 4896: 4865: 6172:", which my dictionary does, but Knowledge doesn't. — 5708:

or elsewhere. Maybe you could elaborate here as well?

4479: 4397: 4228: 4085: 3952: 3829: 3756: 3563:

is transformed to, and the second row expresses where

3508: 3378: 3363: 2816:

can be merged into a single sentence in this article.

939: 884: 855: 826: 771: 646: 605: 578: 551: 442: 357:

looked in it since then, and ought to have remembered.

5545:

Or, perhaps that a matrix could be used to map the {c

5166:

What functions ψ can be expressed by the basis set {φ

5056: 5041:. The operation is then represented by a matrix, say 4636: 4390: 4222: 4079: 3946: 3823: 3750: 3502: 3372: 1986:

Matrices with only one row or column are also called

1953:

Matrices with only one row or column are also called

976: 933: 820: 752: 689: 668:{\displaystyle {\begin{bmatrix}0&1\end{bmatrix}}} 640: 545: 430: 204: 2669:

Subsequent comments should be made in a new section.

378:

Element is overwhelmingly most common -- see below.

5741:magnitude of exotic characters can be distracting. 5542:

under appropriate circumstance could be used for χ?

5209:These and other issues are topics in the theory of 2607:From many books, matrix representation of groups. 5228:where the basis functions are used to approximate 5151: 4775: 4566: 4354: 4201: 4058: 3925: 3802: 3547: 3467:. A major application of matrices is to represent 3421: 999: 963:{\displaystyle {\begin{bmatrix}b\\d\end{bmatrix}}} 962: 919: 806: 734: 667: 626: 477: 225: 3240:"A major application of matrices is to represent 5240:. The basis functions in such methods often are 1317:What I suggest is that you put into a footnote 535:The parallelogram in the figure is obtained by 5910:we are opposed to every single thing you wrote 5232:with matrix equations, for example, using the 2197:(matrix multiplication) is impossible, while 4980:Matrices have application in connection with 2989:should probably just be copy and pasted into 2672:A summary of the conclusions reached follows. 2132:(italics font) can be viewed as a 1x1 matrix 1980:Matrices with only one entry are also called 1947:Matrices with only one entry are also called 139:My suggestion would be "rectangular array of 8: 5527:Or, perhaps that the same kind of expansion 2010:) are not matrices and are sometimes called 351:Introductiong to the theory of linear spaces 6194:http://en.wikipedia.org/Lemma_(mathematics) 4373: 2712:This is a merger proposal was suggested by 414:I believe what the article should say is: 5236:to introduce powerful techniques like the 5224:These ideas have important application to 3433:An example of a matrix with six elements. 2790:be mentioned (but quite rightly are not). 226:{\displaystyle {\underset {m\times n}{A}}} 6001:Frankly I don't know where to start with 5205:corresponding to an operation determined? 5139: 5131: 5123: 5113: 5100: 5084: 5074: 5061: 5055: 4641: 4635: 4478: 4396: 4389: 4223: 4221: 4080: 4078: 3947: 3945: 3824: 3822: 3751: 3749: 3647:Sorry. I was responding to this, above: 3503: 3501: 3491:, this can be represented by the matrix, 3373: 3371: 991: 981: 975: 934: 932: 879: 850: 821: 819: 766: 757: 751: 726: 716: 706: 688: 641: 639: 600: 573: 546: 544: 437: 429: 205: 203: 3337:down to the point where the fundamental 2124:On second thought, I am not sure that a 1560:You've already put the reader to sleep. 6384: 5263: 5132: 3615:From Knowledge Manual of Style (layout) 3008:Reply to "This is Great" section above. 2483:What is the perspective supposed to be? 1850:Array = tensor? Column matrix = vector? 1098:" and was first used in mathematics by 802: 473: 2163:For instance, the product of a scalar 1641:What you are defining is a linear map 492:can be viewed as the transform of the 292:matrix multiplication#Ordinary product 290:of the product matrix, in the section 274:E.g., in the beginning of the section 44:Do not edit the contents of this page. 5986:which in my opinion should not fail. 5919:the oppression inherent in the system 5307:. Cambridge University Press. pp. 94 5274:Fourier analysis and its applications 4805:Characteristic polynomial's variable? 3738:Wikimedia matrix reflects a LaTeX bug 3670:f(x) = 4x." looks fairly good to me. 1530:for her perseverance in this matter. 7: 6334:Nice tool. (For lexical research.) 2663:The following discussion is closed. 1936:Arrays with more than two dimensions 6307:(the last one has a weird history). 5304:The nature of mathematical modeling 3028:In most cases matrices are used to 2864:might be merged (or redirected) to 735:{\displaystyle xA=(A^{T}x^{T})^{T}} 5117: 5114: 5071: 742:,we could equally have transposed 24: 5011:} that can be viewed as a column 396:Interpretation as a parallelogram 6320:Very cool: thanks for the link. 5140: 5124: 4992:} can be used to form functions 2972:The discussion above is closed. 1065:But I didn't want to go so far. 29: 6190:http://en.wikipedia.org/Stadium 5033:and a new correspondence χ → {d 2100:We use it without explaining it 1321:what you know to be the facts: 296:matrix product#Hadamard product 278:the ordinary definition of the 6394:reviews matrix multiplication. 6179:13:50, 27 September 2012 (UTC) 6162:12:53, 27 September 2012 (UTC) 5921:. What about the other points? 3356:Suggestion for first paragraph 723: 699: 1: 5969:21:01, 27 February 2012 (UTC) 5950:01:32, 24 February 2012 (UTC) 5935:17:49, 23 February 2012 (UTC) 5900:16:40, 23 February 2012 (UTC) 5871:16:43, 23 February 2012 (UTC) 5852:16:36, 23 February 2012 (UTC) 5834:16:26, 23 February 2012 (UTC) 5780:Functions and function spaces 5751:18:36, 21 February 2012 (UTC) 5736:18:23, 21 February 2012 (UTC) 5718:17:21, 21 February 2012 (UTC) 5663:15:59, 21 February 2012 (UTC) 5641:18:48, 21 February 2012 (UTC) 5626:18:40, 21 February 2012 (UTC) 5611:15:42, 21 February 2012 (UTC) 5596:12:14, 21 February 2012 (UTC) 5572:03:56, 21 February 2012 (UTC) 5501:20:20, 19 February 2012 (UTC) 5485:18:04, 19 February 2012 (UTC) 5470:17:24, 19 February 2012 (UTC) 5447:04:31, 21 February 2012 (UTC) 5433:04:26, 21 February 2012 (UTC) 5413:21:02, 19 February 2012 (UTC) 5394:17:04, 19 February 2012 (UTC) 4975:Functions and function spaces 4966:17:04, 19 February 2012 (UTC) 4948:10:16, 19 February 2012 (UTC) 4931:04:07, 19 February 2012 (UTC) 4912:02:41, 19 February 2012 (UTC) 4890:02:24, 19 February 2012 (UTC) 4857:Functions and function spaces 3559:The first row expresses what 2193:is a 3x1 column vector, then 1499:"it seems likely that" -: --> 6365:reliable secondary resources 6330:16:17, 24 October 2012 (UTC) 6315:15:45, 24 October 2012 (UTC) 6262:15:19, 24 October 2012 (UTC) 6226:13:57, 24 October 2012 (UTC) 6206:21:34, 23 October 2012 (UTC) 5791:on a function into a matrix 5301:Neil A. Gershenfeld (1999). 4871:section on infinite matrices 4851:12:14, 17 January 2012 (UTC) 4820:11:03, 17 January 2012 (UTC) 3366: 1854:I am not sure that the term 1039:19:59, 3 February 2011 (UTC) 1023:19:38, 3 February 2011 (UTC) 370:22:33, 6 November 2009 (UTC) 341:21:53, 5 November 2009 (UTC) 257:13:14, 18 January 2011 (UTC) 172:13:11, 18 January 2011 (UTC) 157:05:30, 18 January 2011 (UTC) 6377:20:12, 7 January 2013 (UTC) 6344:10:54, 7 January 2013 (UTC) 6302:lemmata vs lemmas vs lemmae 6277:10:43, 7 January 2013 (UTC) 5771:05:49, 13 August 2012 (UTC) 5668:Rschwieb: You've said that 5248:, or in one dimension, the 5037:} to the new column vector 2465:Serge Lang "Linear Algebra" 2008:rectangular parallelepipeds 6420: 4982:generalized Fourier series 2516:21:07, 30 April 2011 (UTC) 2498:21:00, 30 April 2011 (UTC) 2458:20:09, 30 April 2011 (UTC) 2414:15:29, 29 April 2011 (UTC) 2394:12:41, 29 April 2011 (UTC) 2377:14:17, 29 April 2011 (UTC) 2358:12:12, 29 April 2011 (UTC) 2345:12:09, 29 April 2011 (UTC) 2324:12:10, 29 April 2011 (UTC) 2312:12:01, 29 April 2011 (UTC) 2286:11:55, 29 April 2011 (UTC) 2264:11:53, 29 April 2011 (UTC) 2246:11:18, 29 April 2011 (UTC) 2224:11:09, 29 April 2011 (UTC) 2144:seem to be incompatible. 2115:14:36, 30 April 2011 (UTC) 2081:12:41, 29 April 2011 (UTC) 2058:08:30, 29 April 2011 (UTC) 2043:16:45, 28 April 2011 (UTC) 2030:16:45, 28 April 2011 (UTC) 1931:08:28, 29 April 2011 (UTC) 1916:01:00, 29 April 2011 (UTC) 1893:18:50, 28 April 2011 (UTC) 1872:16:18, 28 April 2011 (UTC) 1803:15:01, 29 April 2011 (UTC) 1790:14:07, 29 April 2011 (UTC) 1755:13:48, 28 April 2011 (UTC) 1729:17:41, 27 April 2011 (UTC) 1715:17:04, 27 April 2011 (UTC) 1690:16:00, 27 April 2011 (UTC) 1670:15:28, 27 April 2011 (UTC) 1654:13:01, 27 April 2011 (UTC) 1636:12:10, 27 April 2011 (UTC) 1616:09:56, 27 April 2011 (UTC) 1597:09:29, 26 April 2011 (UTC) 1580:08:52, 26 April 2011 (UTC) 1540:11:22, 16 April 2011 (UTC) 1509:18:49, 12 April 2011 (UTC) 1500:"it is likely that" -: --> 1493:18:47, 12 April 2011 (UTC) 1475:18:20, 12 April 2011 (UTC) 1457:21:19, 11 April 2011 (UTC) 1439:19:29, 11 April 2011 (UTC) 1358:19:08, 11 April 2011 (UTC) 1280:17:31, 11 April 2011 (UTC) 1228:14:37, 11 April 2011 (UTC) 1188:09:25, 11 April 2011 (UTC) 1170:02:15, 11 April 2011 (UTC) 1155:17:55, 10 April 2011 (UTC) 1122:14:30, 10 April 2011 (UTC) 1000:{\displaystyle A^{T}x^{T}} 388:19:24, 30 April 2011 (UTC) 191:17:04, 27 March 2009 (UTC) 181:Can anybody confirm this? 131:19:20, 16 March 2009 (UTC) 113:07:23, 16 March 2009 (UTC) 98:16:54, 15 March 2009 (UTC) 5184:What meaning attaches to 5170:}? This topic is that of 4804: 4799:08:44, 16 June 2011 (UTC) 4618:00:05, 16 June 2011 (UTC) 4591:00:08, 16 June 2011 (UTC) 3114:We can say what a matrix 2596:topics commonly mentioned 1498:"it could be that" -: --> 1090:The term "matrix" is the 1075:14:12, 3 April 2011 (UTC) 500:with vertices at (0,0), ( 18:Talk:Matrix (mathematics) 6144:"matrixes" vs "matrices" 6138:13:57, 7 June 2012 (UTC) 6123:12:21, 7 June 2012 (UTC) 6108:18:42, 6 June 2012 (UTC) 6092:10:28, 6 June 2012 (UTC) 6077:13:08, 5 June 2012 (UTC) 6064:12:24, 5 June 2012 (UTC) 6047:02:15, 5 June 2012 (UTC) 6033:22:21, 4 June 2012 (UTC) 5996:21:22, 4 June 2012 (UTC) 5908:@Brews: Nobody has said 3732:12:00, 1 June 2011 (UTC) 3714:04:12, 1 June 2011 (UTC) 3695:19:40, 31 May 2011 (UTC) 3680:19:40, 31 May 2011 (UTC) 3664:14:01, 31 May 2011 (UTC) 3642:12:53, 31 May 2011 (UTC) 3629:12:31, 31 May 2011 (UTC) 3610:00:50, 29 May 2011 (UTC) 3594:20:42, 28 May 2011 (UTC) 3578:17:41, 28 May 2011 (UTC) 3351:22:06, 30 May 2011 (UTC) 3314:17:18, 28 May 2011 (UTC) 3296:17:08, 28 May 2011 (UTC) 3282:15:45, 28 May 2011 (UTC) 3266:15:31, 28 May 2011 (UTC) 3229:09:34, 29 May 2011 (UTC) 3133:14:16, 28 May 2011 (UTC) 3106:13:59, 28 May 2011 (UTC) 3087:06:47, 28 May 2011 (UTC) 3071:13:04, 26 May 2011 (UTC) 3056:12:34, 26 May 2011 (UTC) 3041:12:30, 26 May 2011 (UTC) 3023:12:17, 26 May 2011 (UTC) 3003:23:05, 25 May 2011 (UTC) 2974:Please do not modify it. 2967:06:09, 20 May 2011 (UTC) 2949:17:07, 19 May 2011 (UTC) 2932:14:09, 19 May 2011 (UTC) 2878:13:53, 19 May 2011 (UTC) 2848:13:18, 19 May 2011 (UTC) 2826:13:05, 19 May 2011 (UTC) 2802:15:10, 19 May 2011 (UTC) 2781:12:21, 19 May 2011 (UTC) 2763:12:15, 19 May 2011 (UTC) 2744:12:14, 19 May 2011 (UTC) 2729:08:58, 19 May 2011 (UTC) 2666:Please do not modify it. 2584:23:11, 25 May 2011 (UTC) 2438:23:36, 25 May 2011 (UTC) 1045:Change to multiplication 539:each of the row vectors 5174:of the basis functions. 2914:column and row matrices 2654:17:51, 1 May 2011 (UTC) 2627:03:19, 1 May 2011 (UTC) 2569:02:30, 1 May 2011 (UTC) 2477:17:47, 1 May 2011 (UTC) 1844:13:47, 3 May 2011 (UTC) 1829:02:35, 3 May 2011 (UTC) 5230:differential equations 5153: 4833:is the polynomial and 4777: 4568: 4567:{\displaystyle \left.} 4356: 4203: 4060: 3927: 3804: 3549: 3469:linear transformations 3423: 3323:linear transformations 3242:linear transformations 2443:Element or entry redux 1237:says in the entry for 1100:James Joseph Sylvester 1063: 1055: 1001: 964: 921: 808: 736: 669: 628: 479: 269:Category:Matrix theory 227: 6292:matrices vs. matrixes 5238:finite element method 5198:How are the elements 5154: 4895:I have to agree with 4778: 4776:{\displaystyle \left} 4569: 4357: 4204: 4061: 3928: 3805: 3550: 3424: 3341:of matrices is lost. 2158:matrix multiplication 2154:scalar multiplication 2120:A scalar is a matrix? 1059: 1051: 1002: 965: 922: 809: 737: 670: 629: 480: 346:Actually, looking in 228: 42:of past discussions. 6299:formulae vs formulas 5859:Matrix (mathematics) 5785:Matrix (mathematics) 5250:triangular functions 5054: 4634: 4388: 4220: 4077: 3944: 3821: 3748: 3500: 3370: 3033:should be mentioned. 2707:Matrix (mathematics) 2699:Matrix (mathematics) 1393:with quotes such as: 1057:would be better as: 974: 931: 818: 750: 687: 638: 543: 428: 202: 3451:, or less commonly 2987:Single-entry matrix 2862:single-entry matrix 2836:Single-entry matrix 2814:single-entry matrix 2703:Single-entry matrix 2695:Single-entry matrix 2693:I propose to merge 2302:Agreed, thank you. 6305:stadia vs stadiums 6296:minima vs minimums 5226:numerical analysis 5149: 4773: 4767: 4564: 4554: 4472: 4376: 4352: 4343: 4199: 4190: 4056: 4047: 3923: 3914: 3800: 3791: 3545: 3536: 3419: 3413: 2714:Michael P. Barnett 2619:Michael P. Barnett 2561:Michael P. Barnett 2490:Michael P. Barnett 2450:Michael P. Barnett 1906:) called tensors. 1290: 997: 960: 954: 917: 911: 870: 841: 804: 803: 796: 732: 665: 659: 624: 618: 591: 564: 475: 474: 467: 380:Michael P. Barnett 223: 221: 6026: 5845: 5494: 5463: 5188:basis functions ( 5145: 5122: 5112: 5107: 4905: 4864:was deleted with 4621: 4604:comment added by 4477: 4475: 3437: 3436: 2905:identity matrices 2902:diagonal matrices 2676:The proposal was 2525:with elements in 2236:Kronecker product 2204:(matrix mult.) = 1570:comment added by 1288: 309:is a redirect to 300:entrywise product 206: 77: 76: 54: 53: 48:current talk page 6411: 6404: 6401: 6395: 6389: 6022: 5841: 5813: 5694: 5541: 5520: 5490: 5459: 5323: 5321: 5298: 5292: 5291: 5268: 5158: 5156: 5155: 5150: 5144: 5143: 5135: 5127: 5121: 5120: 5111: 5106: 5105: 5104: 5095: 5094: 5079: 5078: 5066: 5065: 5032: 5006: 4921:articles on WP. 4901: 4782: 4780: 4779: 4774: 4772: 4768: 4620: 4598: 4573: 4571: 4570: 4565: 4560: 4556: 4555: 4476: 4474: 4473: 4361: 4359: 4358: 4353: 4348: 4347: 4208: 4206: 4205: 4200: 4195: 4194: 4065: 4063: 4062: 4057: 4052: 4051: 3932: 3930: 3929: 3924: 3919: 3918: 3809: 3807: 3806: 3801: 3796: 3795: 3706:Marc van Leeuwen 3566: 3562: 3554: 3552: 3551: 3546: 3541: 3540: 3490: 3486: 3482: 3478: 3428: 3426: 3425: 3420: 3418: 3417: 3364: 3221:Marc van Leeuwen 3218: 3194: 3182: 3154: 3150: 3079:Marc van Leeuwen 3013:the lay reader. 2938:List of matrices 2920:List of matrices 2893:List of matrices 2866:list of matrices 2858:list of matrices 2794: 2755: 2668: 2278:Marc van Leeuwen 2167:by a 3x3 matrix 1880:array data types 1858:is a synonym of 1782:Marc van Leeuwen 1582: 1532:Marc van Leeuwen 1180:Marc van Leeuwen 1080:Spurious history 1006: 1004: 1003: 998: 996: 995: 986: 985: 969: 967: 966: 961: 959: 958: 926: 924: 923: 918: 916: 915: 875: 874: 846: 845: 813: 811: 810: 805: 801: 800: 762: 761: 741: 739: 738: 733: 731: 730: 721: 720: 711: 710: 683:Note that since 674: 672: 671: 666: 664: 663: 633: 631: 630: 625: 623: 622: 596: 595: 569: 568: 488:then the matrix 484: 482: 481: 476: 472: 471: 421:is a 2×2 matrix 262:Entry or element 232: 230: 229: 224: 222: 220: 68: 56: 55: 33: 32: 26: 6419: 6418: 6414: 6413: 6412: 6410: 6409: 6408: 6407: 6402: 6398: 6390: 6386: 6146: 6031: 6003:your definition 5983: 5850: 5811: 5803: 5799: 5797: 5763:Byronchen150199 5692: 5684: 5680: 5678: 5552: 5548: 5540: 5536: 5532: 5528: 5519: 5515: 5511: 5507: 5499: 5468: 5327: 5326: 5318: 5300: 5299: 5295: 5288: 5270: 5269: 5265: 5234:Galerkin method 5211:function spaces 5203: 5186:infinitely many 5180: 5169: 5096: 5080: 5070: 5057: 5052: 5051: 5036: 5031: 5027: 5023: 5019: 5010: 5005: 5001: 4997: 4993: 4991: 4988:of functions {φ 4910: 4859: 4810:I'm afraid. :) 4807: 4766: 4765: 4760: 4755: 4750: 4745: 4740: 4735: 4730: 4725: 4720: 4715: 4710: 4704: 4703: 4698: 4693: 4688: 4683: 4678: 4673: 4668: 4663: 4658: 4653: 4648: 4637: 4632: 4631: 4622: 4599: 4553: 4552: 4547: 4542: 4537: 4532: 4527: 4522: 4516: 4515: 4510: 4505: 4500: 4495: 4490: 4485: 4471: 4470: 4465: 4460: 4455: 4450: 4445: 4440: 4434: 4433: 4428: 4423: 4418: 4413: 4408: 4403: 4395: 4391: 4386: 4385: 4342: 4341: 4336: 4331: 4326: 4321: 4316: 4311: 4306: 4301: 4296: 4291: 4285: 4284: 4279: 4274: 4269: 4264: 4259: 4254: 4249: 4244: 4239: 4234: 4224: 4218: 4217: 4189: 4188: 4183: 4178: 4173: 4168: 4163: 4158: 4153: 4148: 4143: 4137: 4136: 4131: 4126: 4121: 4116: 4111: 4106: 4101: 4096: 4091: 4081: 4075: 4074: 4046: 4045: 4040: 4035: 4030: 4025: 4020: 4015: 4010: 4005: 3999: 3998: 3993: 3988: 3983: 3978: 3973: 3968: 3963: 3958: 3948: 3942: 3941: 3913: 3912: 3907: 3902: 3897: 3892: 3887: 3882: 3877: 3871: 3870: 3865: 3860: 3855: 3850: 3845: 3840: 3835: 3825: 3819: 3818: 3790: 3789: 3784: 3779: 3773: 3772: 3767: 3762: 3752: 3746: 3745: 3740: 3687:Jakob.scholbach 3672:Jakob.scholbach 3617: 3586:Jakob.scholbach 3568: 3564: 3560: 3535: 3534: 3529: 3520: 3519: 3514: 3504: 3498: 3497: 3488: 3484: 3480: 3476: 3412: 3411: 3406: 3401: 3395: 3394: 3389: 3384: 3374: 3368: 3367: 3358: 3306:Jakob.scholbach 3217: 3196: 3184: 3180: 3164: 3152: 3142: 3010: 2983: 2978: 2977: 2908:sparse matrices 2899:square matrices 2870:Jakob.scholbach 2860:. If anything, 2856:something like 2792: 2753: 2691: 2689:Merger proposal 2686: 2664: 2646:Jakob.scholbach 2598: 2538: 2485: 2469:Jakob.scholbach 2445: 2421: 2234:(or rather the 2122: 1938: 1885:Jakob.scholbach 1852: 1565: 1564:their nature? 1552: 1082: 1047: 1031:Jakob.scholbach 987: 977: 972: 971: 970:and calculated 953: 952: 946: 945: 935: 929: 928: 910: 909: 897: 896: 880: 869: 868: 862: 861: 851: 840: 839: 833: 832: 822: 816: 815: 795: 794: 789: 783: 782: 777: 767: 753: 748: 747: 722: 712: 702: 685: 684: 658: 657: 652: 642: 636: 635: 617: 616: 611: 601: 590: 589: 584: 574: 563: 562: 557: 547: 541: 540: 466: 465: 460: 454: 453: 448: 438: 426: 425: 398: 264: 210: 200: 199: 183:Jakob.scholbach 179: 105:Jakob.scholbach 82: 64: 30: 22: 21: 20: 12: 11: 5: 6417: 6415: 6406: 6405: 6396: 6383: 6382: 6381: 6380: 6379: 6361: 6357: 6347: 6346: 6332: 6288: 6287: 6286: 6285: 6284: 6283: 6282: 6281: 6280: 6279: 6249: 6239: 6238: 6237: 6236: 6235: 6234: 6233: 6232: 6211: 6210: 6209: 6208: 6182: 6181: 6145: 6142: 6141: 6140: 6111: 6110: 6080: 6079: 6052: 6051: 6050: 6049: 6027: 6024:JohnBlackburne 6007:one definition 5982: 5979: 5978: 5977: 5976: 5975: 5974: 5973: 5972: 5971: 5922: 5903: 5902: 5876: 5875: 5874: 5873: 5846: 5843:JohnBlackburne 5822:ghost of Hardy 5809: 5801: 5795: 5756: 5755: 5754: 5753: 5738: 5697: 5696: 5690: 5682: 5676: 5648: 5647: 5646: 5645: 5644: 5643: 5581: 5580: 5579: 5578: 5577: 5576: 5575: 5574: 5560: 5557: 5554: 5550: 5546: 5543: 5538: 5534: 5530: 5525: 5522: 5517: 5513: 5509: 5495: 5492:JohnBlackburne 5464: 5461:JohnBlackburne 5455: 5454: 5453: 5452: 5451: 5450: 5449: 5435: 5403: 5400: 5375: 5374: 5373: 5370: 5364: 5363: 5359: 5355: 5350: 5349: 5348: 5347: 5337: 5336: 5325: 5324: 5316: 5293: 5286: 5262: 5261: 5260: 5259: 5221:and the like. 5207: 5206: 5201: 5196: 5193: 5182: 5178: 5175: 5167: 5160: 5159: 5148: 5142: 5138: 5134: 5130: 5126: 5119: 5116: 5110: 5103: 5099: 5093: 5090: 5087: 5083: 5077: 5073: 5069: 5064: 5060: 5034: 5029: 5025: 5021: 5008: 5003: 4999: 4995: 4989: 4978: 4977: 4971: 4970: 4969: 4968: 4951: 4950: 4936: 4935: 4934: 4933: 4915: 4914: 4906: 4903:JohnBlackburne 4858: 4855: 4854: 4853: 4806: 4803: 4802: 4801: 4787: 4786: 4785: 4784: 4783: 4771: 4764: 4761: 4759: 4756: 4754: 4751: 4749: 4746: 4744: 4741: 4739: 4736: 4734: 4731: 4729: 4726: 4724: 4721: 4719: 4716: 4714: 4711: 4709: 4706: 4705: 4702: 4699: 4697: 4694: 4692: 4689: 4687: 4684: 4682: 4679: 4677: 4674: 4672: 4669: 4667: 4664: 4662: 4659: 4657: 4654: 4652: 4649: 4647: 4644: 4643: 4640: 4597: 4595: 4579: 4577: 4576: 4575: 4574: 4563: 4559: 4551: 4548: 4546: 4543: 4541: 4538: 4536: 4533: 4531: 4528: 4526: 4523: 4521: 4518: 4517: 4514: 4511: 4509: 4506: 4504: 4501: 4499: 4496: 4494: 4491: 4489: 4486: 4484: 4481: 4480: 4469: 4466: 4464: 4461: 4459: 4456: 4454: 4451: 4449: 4446: 4444: 4441: 4439: 4436: 4435: 4432: 4429: 4427: 4424: 4422: 4419: 4417: 4414: 4412: 4409: 4407: 4404: 4402: 4399: 4398: 4394: 4380: 4379: 4378: 4377: 4367: 4365: 4364: 4363: 4362: 4351: 4346: 4340: 4337: 4335: 4332: 4330: 4327: 4325: 4322: 4320: 4317: 4315: 4312: 4310: 4307: 4305: 4302: 4300: 4297: 4295: 4292: 4290: 4287: 4286: 4283: 4280: 4278: 4275: 4273: 4270: 4268: 4265: 4263: 4260: 4258: 4255: 4253: 4250: 4248: 4245: 4243: 4240: 4238: 4235: 4233: 4230: 4229: 4227: 4212: 4211: 4210: 4209: 4198: 4193: 4187: 4184: 4182: 4179: 4177: 4174: 4172: 4169: 4167: 4164: 4162: 4159: 4157: 4154: 4152: 4149: 4147: 4144: 4142: 4139: 4138: 4135: 4132: 4130: 4127: 4125: 4122: 4120: 4117: 4115: 4112: 4110: 4107: 4105: 4102: 4100: 4097: 4095: 4092: 4090: 4087: 4086: 4084: 4069: 4068: 4067: 4066: 4055: 4050: 4044: 4041: 4039: 4036: 4034: 4031: 4029: 4026: 4024: 4021: 4019: 4016: 4014: 4011: 4009: 4006: 4004: 4001: 4000: 3997: 3994: 3992: 3989: 3987: 3984: 3982: 3979: 3977: 3974: 3972: 3969: 3967: 3964: 3962: 3959: 3957: 3954: 3953: 3951: 3936: 3935: 3934: 3933: 3922: 3917: 3911: 3908: 3906: 3903: 3901: 3898: 3896: 3893: 3891: 3888: 3886: 3883: 3881: 3878: 3876: 3873: 3872: 3869: 3866: 3864: 3861: 3859: 3856: 3854: 3851: 3849: 3846: 3844: 3841: 3839: 3836: 3834: 3831: 3830: 3828: 3813: 3812: 3811: 3810: 3799: 3794: 3788: 3785: 3783: 3780: 3778: 3775: 3774: 3771: 3768: 3766: 3763: 3761: 3758: 3757: 3755: 3739: 3736: 3735: 3734: 3719: 3718: 3717: 3716: 3698: 3697: 3682: 3645: 3644: 3616: 3613: 3597: 3596: 3558: 3557: 3556: 3555: 3544: 3539: 3533: 3530: 3528: 3525: 3522: 3521: 3518: 3515: 3513: 3510: 3509: 3507: 3435: 3434: 3430: 3429: 3416: 3410: 3407: 3405: 3402: 3400: 3397: 3396: 3393: 3390: 3388: 3385: 3383: 3380: 3379: 3377: 3362: 3357: 3354: 3331:linear algebra 3319: 3318: 3317: 3316: 3299: 3298: 3269: 3268: 3255: 3254: 3253: 3250:linear algebra 3234: 3233: 3232: 3231: 3209: 3172: 3090: 3089: 3044: 3043: 3009: 3006: 2982: 2979: 2971: 2970: 2969: 2916: 2915: 2912: 2911:block matrices 2909: 2906: 2903: 2900: 2853: 2852: 2851: 2850: 2829: 2828: 2809: 2808: 2807: 2806: 2805: 2804: 2793:Sławomir Biały 2766: 2765: 2754:Sławomir Biały 2747: 2746: 2690: 2687: 2685: 2684: 2683: 2682: 2681: 2659: 2658: 2657: 2656: 2638: 2597: 2594: 2593: 2592: 2591: 2590: 2589: 2588: 2587: 2586: 2556: 2540:is called the 2536: 2521:(III.2) "By a 2484: 2481: 2480: 2479: 2444: 2441: 2420: 2417: 2382: 2381: 2380: 2379: 2361: 2360: 2333: 2332: 2331: 2330: 2329: 2328: 2327: 2326: 2314: 2293: 2292: 2291: 2290: 2289: 2288: 2269: 2268: 2267: 2266: 2249: 2248: 2232:tensor product 2213: 2212: 2211:(scalar mult.) 2121: 2118: 2102: 2101: 2098: 2095: 2088: 2087: 2086: 2085: 2084: 2083: 2063: 2062: 2061: 2060: 2017: 2016: 1992:column vectors 1972: 1971: 1959:column vectors 1937: 1934: 1921: 1920: 1919: 1918: 1896: 1895: 1851: 1848: 1847: 1846: 1816: 1815: 1814: 1813: 1812: 1811: 1810: 1809: 1808: 1807: 1806: 1805: 1764: 1763: 1762: 1761: 1760: 1759: 1758: 1757: 1736: 1735: 1734: 1733: 1732: 1731: 1693: 1692: 1678: 1677: 1657: 1656: 1621: 1620: 1619: 1618: 1600: 1599: 1551: 1548: 1547: 1546: 1545: 1544: 1543: 1542: 1514: 1513: 1512: 1511: 1495: 1478: 1477: 1444: 1443: 1442: 1441: 1423: 1422: 1421: 1420: 1419: 1418: 1410: 1409: 1408: 1407: 1406: 1405: 1397: 1396: 1395: 1394: 1388: 1387: 1386: 1385: 1384: 1383: 1371: 1370: 1369: 1368: 1361: 1360: 1344: 1343: 1342: 1341: 1314: 1313: 1283: 1282: 1267: 1266: 1265: 1264: 1258: 1257: 1253: 1252: 1251: 1250: 1243: 1242: 1191: 1190: 1158: 1157: 1110: 1109: 1108: 1107: 1081: 1078: 1046: 1043: 1042: 1041: 994: 990: 984: 980: 957: 951: 948: 947: 944: 941: 940: 938: 914: 908: 905: 902: 899: 898: 895: 892: 889: 886: 885: 883: 878: 873: 867: 864: 863: 860: 857: 856: 854: 849: 844: 838: 835: 834: 831: 828: 827: 825: 799: 793: 790: 788: 785: 784: 781: 778: 776: 773: 772: 770: 765: 760: 756: 729: 725: 719: 715: 709: 705: 701: 698: 695: 692: 682: 662: 656: 653: 651: 648: 647: 645: 621: 615: 612: 610: 607: 606: 604: 599: 594: 588: 585: 583: 580: 579: 577: 572: 567: 561: 558: 556: 553: 552: 550: 486: 485: 470: 464: 461: 459: 456: 455: 452: 449: 447: 444: 443: 441: 436: 433: 397: 394: 393: 392: 391: 390: 373: 372: 358: 326:matrix entries 298:, the concept 284:matrix product 263: 260: 245: 244: 243: 242: 219: 216: 213: 209: 178: 175: 160: 159: 136: 135: 134: 133: 116: 115: 81: 78: 75: 74: 69: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 6416: 6400: 6397: 6393: 6388: 6385: 6378: 6374: 6370: 6366: 6362: 6358: 6355: 6351: 6350: 6349: 6348: 6345: 6341: 6337: 6336:cerniagigante 6333: 6331: 6327: 6323: 6319: 6318: 6317: 6316: 6313: 6310: 6306: 6303: 6300: 6297: 6293: 6278: 6274: 6270: 6269:cerniagigante 6265: 6264: 6263: 6259: 6255: 6250: 6247: 6246: 6245: 6244: 6243: 6242: 6241: 6240: 6229: 6228: 6227: 6224: 6223: 6217: 6216: 6215: 6214: 6213: 6212: 6207: 6203: 6199: 6198:cerniagigante 6195: 6191: 6186: 6185: 6184: 6183: 6180: 6177: 6176: 6171: 6166: 6165: 6164: 6163: 6159: 6155: 6150: 6143: 6139: 6135: 6131: 6127: 6126: 6125: 6124: 6120: 6116: 6109: 6105: 6101: 6096: 6095: 6094: 6093: 6089: 6085: 6078: 6075: 6072: 6068: 6067: 6066: 6065: 6061: 6057: 6048: 6044: 6040: 6036: 6035: 6034: 6030: 6025: 6020: 6016: 6012: 6008: 6004: 6000: 5999: 5998: 5997: 5993: 5989: 5980: 5970: 5966: 5962: 5958: 5953: 5952: 5951: 5947: 5943: 5938: 5937: 5936: 5932: 5928: 5923: 5920: 5915: 5911: 5907: 5906: 5905: 5904: 5901: 5897: 5893: 5888: 5883: 5878: 5877: 5872: 5868: 5864: 5860: 5855: 5854: 5853: 5849: 5844: 5838: 5837: 5836: 5835: 5831: 5827: 5823: 5817: 5814: 5807: 5794: 5790: 5786: 5781: 5777: 5773: 5772: 5768: 5764: 5760: 5752: 5748: 5744: 5739: 5737: 5733: 5729: 5724: 5721: 5720: 5719: 5715: 5711: 5707: 5706:Hilbert space 5703: 5699: 5698: 5688: 5675: 5671: 5667: 5666: 5665: 5664: 5660: 5656: 5652: 5642: 5638: 5634: 5629: 5628: 5627: 5623: 5619: 5614: 5613: 5612: 5608: 5604: 5600: 5599: 5598: 5597: 5593: 5589: 5585: 5573: 5569: 5565: 5561: 5558: 5555: 5549:} into the {d 5544: 5526: 5523: 5504: 5503: 5502: 5498: 5493: 5488: 5487: 5486: 5482: 5478: 5473: 5472: 5471: 5467: 5462: 5456: 5448: 5444: 5440: 5436: 5434: 5430: 5426: 5421: 5420:column vector 5416: 5415: 5414: 5411: 5408: 5404: 5401: 5397: 5396: 5395: 5391: 5387: 5382: 5381: 5379: 5376: 5371: 5368: 5367: 5366: 5365: 5360: 5356: 5352: 5351: 5345: 5344: 5342: 5339: 5338: 5334: 5333: 5332: 5331: 5319: 5314: 5310: 5306: 5305: 5297: 5294: 5289: 5284: 5280: 5276: 5275: 5267: 5264: 5258: 5255: 5254: 5253: 5251: 5247: 5246:hat functions 5243: 5239: 5235: 5231: 5227: 5222: 5220: 5216: 5215:Hilbert space 5212: 5204: 5197: 5194: 5191: 5187: 5183: 5176: 5173: 5165: 5164: 5163: 5146: 5136: 5128: 5108: 5101: 5097: 5091: 5088: 5085: 5081: 5075: 5067: 5062: 5058: 5050: 5049: 5048: 5046: 5045: 5040: 5017: 5014: 4987: 4983: 4976: 4973: 4972: 4967: 4963: 4959: 4955: 4954: 4953: 4952: 4949: 4946: 4943: 4938: 4937: 4932: 4928: 4924: 4919: 4918: 4917: 4916: 4913: 4909: 4904: 4898: 4894: 4893: 4892: 4891: 4887: 4883: 4877: 4874: 4872: 4867: 4863: 4856: 4852: 4848: 4844: 4840: 4836: 4832: 4828: 4824: 4823: 4822: 4821: 4817: 4813: 4812:91.177.218.85 4800: 4796: 4792: 4788: 4769: 4762: 4757: 4752: 4747: 4742: 4737: 4732: 4727: 4722: 4717: 4712: 4707: 4700: 4695: 4690: 4685: 4680: 4675: 4670: 4665: 4660: 4655: 4650: 4645: 4638: 4630: 4629: 4628: 4627: 4624: 4623: 4619: 4615: 4611: 4607: 4603: 4596: 4593: 4592: 4588: 4584: 4580: 4561: 4557: 4549: 4544: 4539: 4534: 4529: 4524: 4519: 4512: 4507: 4502: 4497: 4492: 4487: 4482: 4467: 4462: 4457: 4452: 4447: 4442: 4437: 4430: 4425: 4420: 4415: 4410: 4405: 4400: 4392: 4384: 4383: 4382: 4381: 4372: 4371: 4370: 4369: 4368: 4349: 4344: 4338: 4333: 4328: 4323: 4318: 4313: 4308: 4303: 4298: 4293: 4288: 4281: 4276: 4271: 4266: 4261: 4256: 4251: 4246: 4241: 4236: 4231: 4225: 4216: 4215: 4214: 4213: 4196: 4191: 4185: 4180: 4175: 4170: 4165: 4160: 4155: 4150: 4145: 4140: 4133: 4128: 4123: 4118: 4113: 4108: 4103: 4098: 4093: 4088: 4082: 4073: 4072: 4071: 4070: 4053: 4048: 4042: 4037: 4032: 4027: 4022: 4017: 4012: 4007: 4002: 3995: 3990: 3985: 3980: 3975: 3970: 3965: 3960: 3955: 3949: 3940: 3939: 3938: 3937: 3920: 3915: 3909: 3904: 3899: 3894: 3889: 3884: 3879: 3874: 3867: 3862: 3857: 3852: 3847: 3842: 3837: 3832: 3826: 3817: 3816: 3815: 3814: 3797: 3792: 3786: 3781: 3776: 3769: 3764: 3759: 3753: 3744: 3743: 3742: 3741: 3737: 3733: 3729: 3725: 3721: 3720: 3715: 3711: 3707: 3702: 3701: 3700: 3699: 3696: 3692: 3688: 3683: 3681: 3677: 3673: 3668: 3667: 3666: 3665: 3661: 3657: 3651: 3648: 3643: 3640: 3637: 3633: 3632: 3631: 3630: 3626: 3622: 3614: 3612: 3611: 3607: 3603: 3595: 3591: 3587: 3582: 3581: 3580: 3579: 3576: 3573: 3542: 3537: 3531: 3526: 3523: 3516: 3511: 3505: 3496: 3495: 3494: 3493: 3492: 3474: 3470: 3466: 3462: 3458: 3454: 3450: 3446: 3442: 3432: 3431: 3414: 3408: 3403: 3398: 3391: 3386: 3381: 3375: 3365: 3361: 3355: 3353: 3352: 3348: 3344: 3343:137.82.175.12 3340: 3334: 3332: 3328: 3324: 3315: 3311: 3307: 3303: 3302: 3301: 3300: 3297: 3294: 3291: 3286: 3285: 3284: 3283: 3279: 3275: 3267: 3264: 3261: 3256: 3251: 3247: 3243: 3239: 3238: 3236: 3235: 3230: 3226: 3222: 3216: 3212: 3208: 3204: 3200: 3192: 3188: 3179: 3175: 3171: 3167: 3162: 3158: 3149: 3145: 3139: 3138: 3137: 3136: 3135: 3134: 3130: 3126: 3121: 3117: 3112: 3108: 3107: 3103: 3099: 3094: 3088: 3084: 3080: 3075: 3074: 3073: 3072: 3068: 3064: 3058: 3057: 3053: 3049: 3042: 3039: 3036: 3031: 3027: 3026: 3025: 3024: 3020: 3016: 3007: 3005: 3004: 3000: 2996: 2995:137.82.175.12 2992: 2991:Sparse matrix 2988: 2980: 2975: 2968: 2964: 2960: 2956: 2953: 2952: 2951: 2950: 2946: 2942: 2939: 2934: 2933: 2929: 2925: 2921: 2913: 2910: 2907: 2904: 2901: 2898: 2897: 2896: 2894: 2890: 2885: 2880: 2879: 2875: 2871: 2867: 2863: 2859: 2849: 2845: 2841: 2837: 2833: 2832: 2831: 2830: 2827: 2823: 2819: 2815: 2811: 2810: 2803: 2799: 2795: 2789: 2784: 2783: 2782: 2778: 2774: 2770: 2769: 2768: 2767: 2764: 2760: 2756: 2749: 2748: 2745: 2741: 2737: 2733: 2732: 2731: 2730: 2726: 2722: 2717: 2715: 2710: 2708: 2704: 2700: 2696: 2688: 2679: 2675: 2674: 2673: 2670: 2667: 2661: 2660: 2655: 2651: 2647: 2643: 2639: 2635: 2631: 2630: 2629: 2628: 2624: 2620: 2614: 2611: 2608: 2605: 2602: 2595: 2585: 2581: 2577: 2576:137.82.175.12 2572: 2571: 2570: 2566: 2562: 2557: 2554: 2550: 2546: 2543: 2539: 2532: 2528: 2524: 2519: 2518: 2517: 2513: 2509: 2504: 2503: 2502: 2501: 2500: 2499: 2495: 2491: 2482: 2478: 2474: 2470: 2466: 2462: 2461: 2460: 2459: 2455: 2451: 2442: 2440: 2439: 2435: 2431: 2430:137.82.175.12 2426: 2419:This is great 2418: 2416: 2415: 2411: 2407: 2402: 2396: 2395: 2391: 2387: 2378: 2374: 2370: 2365: 2364: 2363: 2362: 2359: 2356: 2353: 2349: 2348: 2347: 2346: 2342: 2338: 2325: 2322: 2319: 2315: 2313: 2309: 2305: 2301: 2300: 2299: 2298: 2297: 2296: 2295: 2294: 2287: 2283: 2279: 2275: 2274: 2273: 2272: 2271: 2270: 2265: 2261: 2257: 2253: 2252: 2251: 2250: 2247: 2244: 2241: 2237: 2233: 2228: 2227: 2226: 2225: 2221: 2217: 2210: 2207: 2203: 2200: 2199: 2198: 2196: 2192: 2187: 2185: 2182: 2178: 2174: 2170: 2166: 2161: 2159: 2155: 2150: 2145: 2143: 2139: 2135: 2131: 2127: 2119: 2117: 2116: 2112: 2108: 2099: 2096: 2093: 2092: 2091: 2082: 2078: 2074: 2069: 2068: 2067: 2066: 2065: 2064: 2059: 2056: 2053: 2049: 2048: 2047: 2046: 2045: 2044: 2040: 2036: 2032: 2031: 2027: 2023: 2015: 2013: 2009: 2005: 2001: 1997: 1993: 1989: 1985: 1983: 1977: 1976: 1975: 1970: 1968: 1964: 1960: 1956: 1952: 1950: 1944: 1943: 1942: 1935: 1933: 1932: 1929: 1926: 1917: 1913: 1909: 1905: 1900: 1899: 1898: 1897: 1894: 1890: 1886: 1881: 1877: 1876: 1875: 1873: 1869: 1865: 1861: 1857: 1849: 1845: 1841: 1837: 1833: 1832: 1831: 1830: 1826: 1822: 1804: 1801: 1798: 1793: 1792: 1791: 1787: 1783: 1778: 1774: 1773: 1772: 1771: 1770: 1769: 1768: 1767: 1766: 1765: 1756: 1752: 1748: 1744: 1743: 1742: 1741: 1740: 1739: 1738: 1737: 1730: 1726: 1722: 1718: 1717: 1716: 1712: 1708: 1704: 1701: 1698:I agree with 1697: 1696: 1695: 1694: 1691: 1688: 1685: 1680: 1679: 1674: 1673: 1672: 1671: 1667: 1663: 1655: 1652: 1649: 1644: 1640: 1639: 1638: 1637: 1633: 1629: 1625: 1617: 1613: 1609: 1604: 1603: 1602: 1601: 1598: 1594: 1590: 1585: 1584: 1583: 1581: 1577: 1573: 1569: 1561: 1558: 1555: 1541: 1537: 1533: 1529: 1525: 1520: 1519: 1518: 1517: 1516: 1515: 1510: 1507: 1504: 1496: 1494: 1490: 1486: 1482: 1481: 1480: 1479: 1476: 1472: 1468: 1464: 1461: 1460: 1459: 1458: 1454: 1450: 1440: 1436: 1432: 1427: 1426: 1425: 1424: 1416: 1415: 1414: 1413: 1412: 1411: 1403: 1402: 1401: 1400: 1399: 1398: 1392: 1391: 1390: 1389: 1380: 1377: 1376: 1375: 1374: 1373: 1372: 1365: 1364: 1363: 1362: 1359: 1355: 1351: 1346: 1345: 1339: 1335: 1332: 1329: 1323: 1322: 1320: 1316: 1315: 1311: 1306: 1302: 1301:hard evidence 1298: 1294: 1285: 1284: 1281: 1277: 1273: 1269: 1268: 1262: 1261: 1260: 1259: 1255: 1254: 1247: 1246: 1245: 1244: 1240: 1236: 1233:Here is what 1232: 1231: 1230: 1229: 1225: 1221: 1217: 1212: 1208: 1204: 1200: 1196: 1189: 1185: 1181: 1177: 1174: 1173: 1172: 1171: 1167: 1163: 1156: 1152: 1148: 1144: 1141: 1138: 1135: 1132: 1129: 1126: 1125: 1124: 1123: 1119: 1115: 1105: 1101: 1097: 1093: 1089: 1088: 1087: 1086: 1085: 1079: 1077: 1076: 1072: 1068: 1062: 1058: 1054: 1050: 1044: 1040: 1036: 1032: 1028: 1027: 1026: 1025:Lottie Price 1024: 1020: 1016: 1012: 1010: 992: 988: 982: 978: 955: 949: 942: 936: 912: 906: 903: 900: 893: 890: 887: 881: 876: 871: 865: 858: 852: 847: 842: 836: 829: 823: 797: 791: 786: 779: 774: 768: 763: 758: 754: 745: 727: 717: 713: 707: 703: 696: 693: 690: 680: 678: 660: 654: 649: 643: 619: 613: 608: 602: 597: 592: 586: 581: 575: 570: 565: 559: 554: 548: 538: 533: 531: 527: 523: 519: 515: 511: 507: 503: 499: 498:parallelogram 495: 491: 468: 462: 457: 450: 445: 439: 434: 431: 424: 423: 422: 420: 415: 412: 409: 405: 401: 395: 389: 385: 381: 377: 376: 375: 374: 371: 367: 363: 359: 356: 352: 349: 345: 344: 343: 342: 338: 334: 330: 327: 323: 318: 316: 312: 308: 303: 301: 297: 293: 289: 285: 281: 277: 272: 270: 261: 259: 258: 254: 250: 240: 236: 217: 214: 211: 207: 197: 196: 195: 194: 193: 192: 188: 184: 176: 174: 173: 169: 165: 158: 154: 150: 146: 142: 138: 137: 132: 128: 124: 120: 119: 118: 117: 114: 110: 106: 102: 101: 100: 99: 95: 91: 87: 79: 73: 70: 67: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 6399: 6387: 6364: 6353: 6289: 6221: 6174: 6151: 6147: 6112: 6081: 6053: 6018: 6014: 6010: 5984: 5955: 5913: 5909: 5818: 5815: 5805: 5792: 5788: 5779: 5775: 5774: 5758: 5757: 5722: 5702:completeness 5701: 5686: 5673: 5650: 5649: 5583: 5582: 5419: 5377: 5340: 5329: 5328: 5308: 5303: 5296: 5278: 5273: 5266: 5256: 5249: 5245: 5223: 5219:Banach space 5208: 5199: 5189: 5185: 5172:completeness 5171: 5161: 5043: 5042: 5038: 5015: 4979: 4974: 4878: 4875: 4862:This section 4860: 4838: 4834: 4830: 4826: 4808: 4606:MathInclined 4600:— Preceding 4594: 4583:MathInclined 4581: 4578: 4366: 3724:Rick Norwood 3656:Rick Norwood 3652: 3649: 3646: 3621:Rick Norwood 3618: 3602:Rick Norwood 3598: 3569: 3464: 3460: 3456: 3452: 3448: 3444: 3438: 3359: 3338: 3335: 3320: 3274:Rick Norwood 3270: 3214: 3210: 3206: 3202: 3198: 3190: 3186: 3177: 3173: 3169: 3165: 3160: 3156: 3147: 3143: 3125:Rick Norwood 3119: 3115: 3113: 3109: 3098:Rick Norwood 3095: 3091: 3063:Rick Norwood 3059: 3048:Rick Norwood 3045: 3029: 3015:Rick Norwood 3011: 2984: 2981:another idea 2973: 2954: 2935: 2917: 2888: 2883: 2881: 2854: 2818:Rick Norwood 2787: 2736:Rick Norwood 2718: 2711: 2692: 2677: 2671: 2665: 2662: 2641: 2633: 2615: 2612: 2609: 2606: 2603: 2599: 2552: 2548: 2544: 2541: 2534: 2530: 2529:..." (where 2526: 2522: 2508:Rick Norwood 2486: 2446: 2424: 2422: 2406:Rick Norwood 2400: 2397: 2383: 2337:Rick Norwood 2334: 2214: 2208: 2205: 2201: 2194: 2190: 2188: 2183: 2180: 2176: 2172: 2168: 2164: 2162: 2146: 2141: 2137: 2133: 2129: 2123: 2103: 2089: 2033: 2018: 2003: 1999: 1979: 1978: 1973: 1946: 1945: 1939: 1922: 1903: 1853: 1817: 1776: 1662:Rick Norwood 1658: 1642: 1628:Rick Norwood 1626: 1622: 1608:71.37.42.176 1572:71.37.42.176 1562: 1559: 1556: 1553: 1523: 1445: 1378: 1337: 1333: 1330: 1327: 1318: 1309: 1304: 1300: 1296: 1292: 1238: 1210: 1206: 1202: 1192: 1159: 1111: 1104:determinants 1083: 1064: 1060: 1056: 1052: 1048: 1015:13.1.100.136 1013: 1008: 743: 681: 676: 675:with matrix 534: 529: 525: 521: 517: 513: 509: 505: 501: 489: 487: 418: 416: 413: 410: 406: 402: 399: 354: 350: 331:in en:wiki? 328: 321: 319: 314: 306: 304: 299: 287: 279: 273: 265: 249:Rick Norwood 246: 238: 234: 180: 164:Rick Norwood 161: 123:131.188.3.21 90:131.188.3.21 85: 83: 65: 43: 37: 6360:acceptable? 5942:Brews ohare 5887:WP:COATRACK 5863:Brews ohare 5826:Brews ohare 5710:Brews ohare 5670:my proposal 5603:Brews ohare 5564:Brews ohare 5477:Brews ohare 5439:Brews ohare 5425:Brews ohare 5386:Brews ohare 4958:Brews ohare 4923:Brews ohare 4882:Brews ohare 4839:det(A-λI)=0 3567:is send to. 3441:mathematics 3195:that sends 3189:} × {1,..., 2834:The entire 2642:summarizing 1566:—Preceding 1328:Author_here 1289:Elizabethan 1207:matr, mater 537:multiplying 494:unit square 324:article on 145:expressions 36:This is an 5981:Definition 5330:Discussion 5317:0521570956 5287:0821847902 5257:References 3457:components 2889:improperly 2887:sometimes 1334:title here 1094:word for " 4986:basis set 4897:this edit 4866:this edit 3030:represent 2955:Withdrawn 2678:Withdrawn 2553:component 2189:Also, if 1589:Gandalf61 1338:page here 1331:year_here 1305:reference 1211:something 814:and x to 149:Kragen2uk 72:Archive 3 66:Archive 2 60:Archive 1 6369:Rschwieb 6322:Rschwieb 6254:Rschwieb 6154:Rschwieb 6100:Rschwieb 6039:Rschwieb 5961:Rschwieb 5927:Rschwieb 5759:Comment: 5743:Rschwieb 5728:Rschwieb 5655:Rschwieb 5633:Rschwieb 4940:covered. 4614:contribs 4602:unsigned 3461:elements 3453:matrixes 3449:matrices 3447:(plural 2959:Muhandes 2924:Paolo.dL 2884:non-zero 2840:Muhandes 2773:Muhandes 2721:Muhandes 2545:-element 2386:Paolo.dL 2304:Paolo.dL 2256:Paolo.dL 2216:Paolo.dL 2175:, where 2107:Paolo.dL 2073:Paolo.dL 2035:Paolo.dL 2022:Paolo.dL 2000:Scalars, 1908:Paolo.dL 1864:Paolo.dL 1836:Paolo.dL 1747:Paolo.dL 1721:Saros136 1707:Paolo.dL 1568:unsigned 1524:in Latin 1449:Wvbailey 1350:Wvbailey 1324:ref: --> 1220:Wvbailey 1216:bullshit 1195:Bullshit 1162:Wvbailey 1114:Wvbailey 524:), and ( 362:JoergenB 348:Shilov's 333:JoergenB 329:anywhere 311:Entrance 288:elements 177:Notation 6222:Quondum 6175:Quondum 6170:minimum 5776:Summary 5651:Comment 5584:Comment 5378:Comment 5341:Comment 3473:vectors 3465:entries 3339:purpose 3327:vectors 3288:school. 3246:vectors 3185:{1,..., 3163:matrix 2734:Agree. 2147:In the 2012:tensors 1996:vectors 1982:scalars 1967:tensors 1963:vectors 1949:scalars 1319:exactly 1293:surmise 1218:. Bill 496:into a 280:entries 241:matrix. 233:for an 141:numbers 39:archive 6115:Nijdam 6084:Nijdam 6056:Nijdam 5988:Nijdam 5242:chosen 5013:vector 4835:p(λ)=0 3489:(-1,0) 3483:, and 3445:matrix 2637:there. 2523:matrix 2401:define 2369:RobHar 2149:MATLAB 2126:scalar 2004:zero-, 1904:always 1860:Tensor 1467:Myrvin 1239:matrix 1205:, fr. 1199:WP:BRD 1142:, and 1067:Myrvin 6029:deeds 5957:time. 5882:WP:OR 5848:deeds 5497:deeds 5466:deeds 5213:like 5192:→ ∞)? 4908:deeds 4843:LutzL 4791:LutzL 3565:(0,1) 3561:(1,0) 3485:(0,1) 3481:(0,1) 3477:(1,0) 3463:, or 3193:} → ℚ 2788:could 2697:into 2549:entry 2542:(i,k) 1856:Array 1506:Adler 1382:hist. 1092:Latin 307:Entry 16:< 6373:talk 6340:talk 6326:talk 6273:talk 6258:talk 6202:talk 6158:talk 6134:talk 6130:Dmcq 6119:talk 6104:talk 6088:talk 6060:talk 6043:talk 6017:and 5992:talk 5965:talk 5946:talk 5931:talk 5896:talk 5892:Dmcq 5867:talk 5830:talk 5767:talk 5747:talk 5732:talk 5714:talk 5659:talk 5637:talk 5622:talk 5618:Dmcq 5607:talk 5592:talk 5588:Dmcq 5568:talk 5529:χ = 5508:ψ = 5481:talk 5443:talk 5429:talk 5390:talk 5313:ISBN 5283:ISBN 5020:χ = 4994:ψ = 4962:talk 4927:talk 4886:talk 4847:talk 4831:p(t) 4816:talk 4795:talk 4610:talk 4587:talk 3728:talk 3710:talk 3691:talk 3676:talk 3660:talk 3625:talk 3606:talk 3590:talk 3443:, a 3347:talk 3310:talk 3278:talk 3225:talk 3205:) ↦ 3129:talk 3102:talk 3083:talk 3067:talk 3052:talk 3019:talk 2999:talk 2963:talk 2945:talk 2941:Bhny 2928:talk 2874:talk 2844:talk 2822:talk 2798:talk 2777:talk 2759:talk 2740:talk 2725:talk 2650:talk 2623:talk 2580:talk 2565:talk 2512:talk 2494:talk 2473:talk 2454:talk 2434:talk 2410:talk 2390:talk 2373:talk 2341:talk 2308:talk 2282:talk 2260:talk 2230:the 2220:talk 2140:and 2111:talk 2077:talk 2039:talk 2026:talk 1912:talk 1889:talk 1868:talk 1840:talk 1825:talk 1821:Bhny 1786:talk 1777:body 1751:talk 1725:talk 1711:talk 1666:talk 1632:talk 1612:talk 1593:talk 1576:talk 1536:talk 1528:Mhym 1503:Hans 1489:talk 1485:Mhym 1471:talk 1453:talk 1435:talk 1431:Mhym 1367:OED: 1354:talk 1310:fact 1297:fact 1276:talk 1272:Mhym 1249:n.). 1224:talk 1203:womb 1184:talk 1166:talk 1151:talk 1147:Mhym 1118:talk 1112:Bill 1096:womb 1071:talk 1035:talk 1019:talk 927:and 634:and 532:). 508:), ( 384:talk 366:talk 355:have 337:talk 253:talk 237:-by- 187:talk 168:talk 153:talk 127:talk 109:talk 94:talk 6354:for 5252:. 5202:m,n 4837:or 3487:to 3479:to 3471:of 3439:In 3329:in 3325:of 3248:in 3244:of 3168:= ( 3153:x,y 3146:= 2 3120:why 2634:are 2551:or 2186:. 1990:or 1988:row 1957:or 1955:row 1643:not 1299:or 1235:OED 746:to 417:If 322:any 315:any 143:or 86:not 6375:) 6342:) 6328:) 6294:, 6275:) 6260:) 6204:) 6192:, 6160:) 6136:) 6121:) 6106:) 6090:) 6062:) 6045:) 6013:, 5994:) 5967:) 5948:) 5933:) 5914:is 5898:) 5869:) 5861:. 5832:) 5812:). 5800:(φ 5796:ij 5769:) 5749:) 5734:) 5716:) 5685:, 5681:(φ 5677:ij 5661:) 5639:) 5624:) 5609:) 5594:) 5570:) 5553:}? 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5518:n 5516:φ 5514:n 5512:c 5510:Σ 5479:( 5441:( 5427:( 5410:R 5407:T 5388:( 5320:. 5290:. 5200:M 5190:n 5179:n 5168:n 5147:. 5141:c 5133:M 5129:= 5125:d 5118:r 5115:o 5109:, 5102:n 5098:c 5092:n 5089:, 5086:m 5082:M 5076:n 5068:= 5063:m 5059:d 5044:M 5039:d 5035:n 5030:n 5028:φ 5026:n 5024:d 5022:Σ 5016:c 5009:n 5004:n 5002:φ 5000:n 4998:c 4996:Σ 4990:n 4960:( 4945:R 4942:T 4925:( 4884:( 4845:( 4827:t 4814:( 4793:( 4770:] 4763:l 4758:k 4753:j 4748:i 4743:h 4738:g 4733:f 4728:e 4723:d 4718:c 4713:b 4708:a 4686:9 4681:8 4676:7 4671:6 4666:5 4661:4 4656:3 4651:2 4646:1 4639:[ 4608:( 4585:( 4562:. 4558:] 4550:n 4545:m 4540:l 4535:k 4530:j 4525:i 4520:h 4488:9 4483:8 4468:g 4463:f 4458:e 4453:d 4448:c 4443:b 4438:a 4431:7 4426:6 4421:5 4416:4 4411:3 4406:2 4401:1 4393:[ 4350:. 4345:] 4339:k 4334:j 4329:i 4324:h 4319:g 4314:f 4309:e 4304:d 4299:c 4294:b 4289:a 4272:9 4267:8 4262:7 4257:6 4252:5 4247:4 4242:3 4237:2 4232:1 4226:[ 4197:. 4192:] 4186:j 4181:i 4176:h 4171:g 4166:f 4161:e 4156:d 4151:c 4146:b 4141:a 4129:9 4124:8 4119:7 4114:6 4109:5 4104:4 4099:3 4094:2 4089:1 4083:[ 4054:. 4049:] 4043:i 4038:h 4033:g 4028:f 4023:e 4018:d 4013:c 4008:b 4003:a 3996:9 3991:8 3986:7 3981:6 3976:5 3971:4 3966:3 3961:2 3956:1 3950:[ 3921:. 3916:] 3910:h 3905:g 3900:f 3895:e 3890:d 3885:c 3880:b 3875:a 3868:8 3863:7 3858:6 3853:5 3848:4 3843:3 3838:2 3833:1 3827:[ 3798:. 3793:] 3787:6 3765:9 3760:1 3754:[ 3726:( 3708:( 3689:( 3674:( 3658:( 3639:R 3636:T 3623:( 3604:( 3588:( 3575:R 3572:T 3543:. 3538:] 3532:0 3527:1 3517:1 3512:0 3506:[ 3415:] 3409:6 3387:9 3382:1 3376:[ 3345:( 3308:( 3293:R 3290:T 3276:( 3263:R 3260:T 3223:( 3215:j 3213:, 3211:i 3207:a 3203:j 3201:, 3199:i 3197:( 3191:n 3187:m 3181:) 3178:j 3176:, 3174:i 3170:a 3166:A 3161:n 3157:m 3148:x 3144:y 3127:( 3100:( 3081:( 3065:( 3050:( 3038:R 3035:T 3017:( 2997:( 2961:( 2943:( 2926:( 2872:( 2842:( 2820:( 2796:( 2775:( 2757:( 2738:( 2723:( 2680:. 2648:( 2621:( 2578:( 2563:( 2547:( 2535:a 2531:K 2527:K 2510:( 2492:( 2471:( 2452:( 2432:( 2408:( 2388:( 2371:( 2355:R 2352:T 2339:( 2321:R 2318:T 2306:( 2280:( 2258:( 2243:R 2240:T 2218:( 2209:s 2206:v 2191:v 2184:I 2181:s 2177:S 2169:M 2165:s 2142:s 2138:s 2134:s 2130:s 2109:( 2075:( 2055:R 2052:T 2037:( 2024:( 2014:. 1984:. 1969:. 1951:. 1928:R 1925:T 1910:( 1902:( 1887:( 1866:( 1838:( 1823:( 1800:R 1797:T 1784:( 1749:( 1723:( 1709:( 1703:R 1700:T 1687:R 1684:T 1664:( 1651:R 1648:T 1630:( 1610:( 1591:( 1574:( 1534:( 1487:( 1469:( 1451:( 1433:( 1352:( 1336:: 1274:( 1241:: 1222:( 1182:( 1164:( 1149:( 1116:( 1106:. 1069:( 1033:( 1017:( 993:T 989:x 983:T 979:A 956:] 950:d 943:b 937:[ 913:] 907:d 904:+ 901:c 894:b 891:+ 888:a 882:[ 877:, 872:] 866:c 859:a 853:[ 848:, 843:] 837:0 830:0 824:[ 798:] 792:d 787:b 780:c 775:a 769:[ 764:= 759:T 755:A 744:A 728:T 724:) 718:T 714:x 708:T 704:A 700:( 697:= 694:A 691:x 677:A 661:] 655:1 650:0 644:[ 620:] 614:1 609:1 603:[ 598:, 593:] 587:0 582:1 576:[ 571:, 566:] 560:0 555:0 549:[ 530:d 528:, 526:c 522:d 518:b 514:c 510:a 506:b 504:, 502:a 490:A 469:] 463:d 458:c 451:b 446:a 440:[ 435:= 432:A 419:A 382:( 364:( 335:( 251:( 239:n 235:m 218:n 212:m 208:A 185:( 166:( 151:( 125:( 107:( 92:( 50:.

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