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970:{\displaystyle {\begin{pmatrix}A_{11}&C_{11}&A_{12}&C_{12}\\-{\overline {C_{11}}}&{\overline {A_{11}}}&-{\overline {C_{12}}}&{\overline {A_{12}}}\\A_{21}&C_{21}&A_{22}&C_{22}\\-{\overline {C_{21}}}&{\overline {A_{21}}}&-{\overline {C_{22}}}&{\overline {A_{22}}}\end{pmatrix}}}
188:
Copuls someone perhaps explain the relationship between these four groups and eucledian geometry? I note that one of them mentions "having a deteminant of 1", so it seems to be something to do withh affine transformations that preserve area. Maybe a few paragraphs pitched at a slightly easier level?
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I think what is meant here is to merge the section on classical groups from the article "matrix groups" into this article, where it would fit better. Perhaps, the title would have to be changed to "Classical group". This would allow us to concentrate on the general properties of matrix (i.e. linear)
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It's hard for me to follow exactly what is being contested, but my take on groups like SL(n,H) is that these are subgroups of SL(4n,R) preserving a pair of anti commuting complex structures. Thus they are quaternionic in the sense that they have a fundamental representation that is quaternionic. I
157:
I strongly oppose merging "Matrix group" and "Classical Lie group". They are very different topics. A matrix group is any group of matrices. It may be finite, it may have coefficients in a finite field, it may be a topologically discrete subgroup of a classical Lie group, etc. A classical Lie
1372:
relevant to the subject, and contains several errors. There are plenty of articles that are complete, correct, and good reading, but will never get a C-class rating because the subject is narrow and will never motivate more than one page of text. (I disagree with this rating policy b t w.)
1517:
The determinant of a quaternionic matrix is defined in this representation as being the ordinary complex determinant of its representative matrix. The non-commutative nature of quaternionic multiplication would, in the quaternionic representation of matrices, be
197:
There is no way this article is class B, so I gave it a "Start" rating. The subject has inspired much classical research, many books, and is the backbone in any course beyond "abstract group theory". It deserves a much more inspired article.
994:, because 8 terms are real by construction and the rest 16 come in 8 conjugated pairs. By applying one real equation to eight complex variables, one can define what is the special linear group of the rank 2 âoverâ quaternions, a subgroup of
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A while ago, I changed the article rating from B-class to start-class (motivated above). Someone now changed this to C-class with the motivation that "article ratings aren't arbitrary". Right, they aren'tâat least they
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the theory of particular symmetries without exaggerating very much. These symmetries are often described by classical groups.) With this in mind, the article is, in my opinion, smack dab start-class, even a
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the "non-commutative nature of quaternionic multiplication" that would be ambiguous; it is the definition of the determinant of a quaternionic matrix that would be ambiguous.
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You are on the right track. I don't have time a t m, but I'll check later in
Rossmann's book. B t w, the matrix representation to use for the quaternions in this context is
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start-class article, but it is not C-class. I will not change the rating back unless others agree with me, and I can live with this as an unusually poor C-class article.
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1262:(real or complex numbers, both are feasible). This case is lucky to admit this possibility. The main question is not quaternions, but generality. Read the section
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An introductory exposition of these forms and their signatures, could explain all terminology and some connections between the groups. A beefier explanation of '
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group is a manifold with group structure; this gives it very particular properties. There are different questions and results that belong to the two topics.
213:. Variants exist; the connoisseur might add the exceptional groups, others relax the determinant = 1 condition to determinant = +/-1, yet others include also
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Most (all actually, and much more) of what I propose could be extracted from §3.1 in Wulf
Rossmann's "Lie Groups, an Introduction through Linear Groups".
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Certainly groups like SL(n,H) are regarded as classical groups. (SL(n,H) has type AII in Cartan's classification if I recall correctly.)
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classical topic that is extremely useful in modern applications. (You can take it to the extreme point that modern theoretical physics
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it, giving it a solid start. I'm merely complaining. I wish I had the time to develop the article, but I don't.
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If it is the construction suggested by YohanN7, then a rhetorical question: has any non-commutative ring a
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1217:. Disclaimer: This, or something similar, is true, can't check this thoroughly a t m, playing poker;).
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1419:. I think it should go elsewhere (or be scrapped) because it isn't in the spirit of classical groups.
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is a column vector representation of a quaternion. With the former expression, you would need to use
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is a completely unnecessary extension of the subject. It is far from the classical considerations.
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Does Incnis Mrsi make unnecessarily aggressive, personalized comments because they are opposed to
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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said: linear maps that preserve the quaternionic structure, but their determinant is calculated
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398:{\displaystyle \scriptstyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}}
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1136:{\displaystyle q={\begin{bmatrix}A&-{\overline {C}}\\C&{\overline {A}}\end{bmatrix}}.}
300:. Articles on particular classical groups do not define them over anything more general than
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639:{\displaystyle q={\begin{bmatrix}A&C\\-{\overline {C}}&{\overline {A}}\end{bmatrix}}}
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1266:(including the headline) until the ârespected by the matrix multiplicationâ words, please.
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This article is one page (at least upon removal of irrelevant stuff) on a fully developed
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non-commutative rings have such representations? There was something wrong with grammar.
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that is relevant to this article, as well as some particular classical groups such as
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as quaternions that have its determinant equal to 1, i.e. unit quaternions. As for
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article, as opposed to descriptions of classical groups, as is presently the case.
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Exactly. With the latter expression you can multiply using matrix multiplication,
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matrix representation over a commutative ring? Of course, not any.
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Of course, I'm not blaming the article's authors. They, at least,
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matrix, of course, has its determinant well-defined and even
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I rewrote the article from scratch but retained the section
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didnât learn specified sources, but the only possibility
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and, consequently, the order of matrix multiplication.
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1323:. That is to say, vector spaces over
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201:In today's terminology there are the
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1468:use of sans-serif in math notation?
312:. How can one define it for, say, a
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1508:Near the end of the section titled
1464:} for transpose due to aversion to
649:representation, as it is suggested
38:It is of interest to the following
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563:that can define quaternions from
115:Knowledge:WikiProject Mathematics
118:Template:WikiProject Mathematics
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255:On the other hand, the section
135:This article has been rated as
1486:forms of polite engagement? --
1:
1292:17:29, 18 February 2014 (UTC)
1276:17:18, 18 February 2014 (UTC)
1254:Not very different from what
1250:16:51, 18 February 2014 (UTC)
1227:18:47, 18 February 2014 (UTC)
1187:17:18, 18 February 2014 (UTC)
1165:15:07, 18 February 2014 (UTC)
1052:17:18, 18 February 2014 (UTC)
1029:12:07, 18 February 2014 (UTC)
109:and see a list of open tasks.
1550:C-Class mathematics articles
1457:{\displaystyle \mathrm {T} }
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1155:(scalars go to the right).
561:CayleyâDickson construction
551:is possible. Is it really?
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1470:Incnis Mrsi
1438:like serif
1411:New version
1387:pretty good
1359:Not C-class
1268:Incnis Mrsi
1179:Incnis Mrsi
1044:Incnis Mrsi
1021:Incnis Mrsi
310:determinant
193:Not B-class
112:Mathematics
103:mathematics
59:Mathematics
1544:Categories
1522:But it is
1298:The cases
304:, such as
1366:shouldn't
1175:transpose
980:and this
288:There is
1205:, where
227:SU(p, q)
209:and the
1440:\mathrm
1436:YohanN7
1421:YohanN7
1398:YohanN7
1343:YohanN7
1219:YohanN7
1157:YohanN7
535:argues
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