673:? The two references given are modern textbooks on electromagnetism, and we know, from electromagnetism, that these really are tensor products, etc. I would be much happier if the reference was devoted to dyadic algebra as such, rather than a one-off physics book. Similarly, a clear statement that "dyadic algebra is like tensor algebra, except that axiom blah blah is not used", or whatever. Otherwise, I see no clear distinction, just a somewhat awkward notation.
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two vectors (or higher order tensors). I'm not a mathematician and I came here because I needed some practical identities regarding outer product. I think it would be useful for nonmathematicians if a definition or a reference of the unsigned multiplication is given, because it is evidently not what it is usually supposed to be (neither scalar multiplication nor dyadic product). Thanks. Eratostene
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multiplication, creating a square matrix that represents the dyadic product of the two original vectors. The only reason for using "column vector" and "row vector" is to make their matrix representations intuitive, so if we're going to change or generalize anything, we should emphasize that "row" and "column" are only important in
457:{\displaystyle \mathbf {v} \otimes \mathbf {u} ={\begin{bmatrix}v_{1}&v_{2}&v_{3}\end{bmatrix}}\otimes {\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}={\begin{bmatrix}v_{1}u_{1}&v_{1}u_{2}&v_{1}u_{3}\\v_{2}u_{1}&v_{2}u_{2}&v_{2}u_{3}\\v_{3}u_{1}&v_{3}u_{2}&v_{3}u_{3}\end{bmatrix}}}
836:
I landed here via Random
Article. I've been trying to figure out what any of the sentences in this article or talk page mean. I have absolutely NO CLUE what this thing is, what it does or if it exists outside the minds of mathematicians. Therefore, I have absolutely NO CLUE if or where it should be
766:
Given that all u, v and w are vectors, I don't understand the unsigned multiplication used in the 4th and 5th identities of the paragraph "Identities". I usually use unsigned multiplication for scalar multiplication (i.e. a scalar times a vector or tensor) and for outer (or dyadic) product between
639:
The notation is a little different and to merge any mention of dyadics or polyadics into an article on tensors would only serve to make that article considerably more confusing. My suggestion is to keep the tensor and polyadics articles completely separate aside from a possible mention of their
476:
Technically, tensor products of vectors (and vectors themselves) are defined without a particular matrix structure in mind (i.e., "row" versus "column" is unimportant), so the question is one of representation only. The ordering of (column) x (row) is chosen to match the intuition of matrix
545:
This is an old discussion, but for the record here's my response. When you say "an outer product of two vectors produces another vector"... which "outer product" are you referring to? I thought the outer product of two vectors produces an order-2 tensor i.e. matrix -
687:
The dyadic product is the tensor product of two 1-vectors/1-forms, whereas the tensor product applies more generally: to tensors of arbitrary order. Given that this distinction can easily be made clear, I withdraw any objection to merging. What I meant by the "full
861:
The introduction distinguishes the "order" and "rank" of the resulting tensor. However, the page on tensors (linked to for both terms) indicates that these terms mean the same thing. So what exactly is meant by "order" and "rank" here?
794:
If you do the dot-product multiplication first in either identity's right-hand side, you see there is no "undefined multiplication", since the remaining operation is multiplication of a vector by a scalar.
746:-- the only two references in dyadics are two modern books on electromagnetism! And we know that in electromagnetism, the usual scalar, vector, tensor products work as usual; there is no distinct notion.
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937:, so to try and fix that a statement of the equivalence between "dyadic = outer = tensor" product in the context of dyadics, including the notation, has been added in
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merged. But given our agreement that it is a "small article for something" and "just a bit confusing", I feel reasonably confident we must agree on merging as well. I
692:" is perhaps also an arcane point, relating to how tensors of different order may be added in the formal algebra, apparenlty not considered in polyadics. —
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An outer product of two vectors produces another vector, while the dyadic product produces a second-order tensor, a matrix. So, no merger in my opinion.
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being used for two distinct concepts), but usage seems IMO to be converging on the usage in this article (at least within
Knowledge). Preferably, the
814:
There is really no need for this article, its just a bit confusing to have a small article for something just because it has more than one name.
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An outer product in fact yields a second-order tensor/matrix as well...In all honesty, I see no reason for this separate article.
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721:"dyadic product" is only a sub-operation of "dyadics", so if there has to be a merger, it has to be the other way around, and
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for that content in the latter page, and it must not be deleted as long as the latter page exists.
724:"dyadic product" is the most used operation of the "dyadics", and deserves an article on its own.
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To call the dyadic product a second-order tensor may not be such a good idea though:
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The indices in the definitions would have to be swapped in that case. --
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The words are not used consistently historically (especially
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I oppose to a merger of "dyadics" into this article, because
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Given the consensus to merge this article into the other
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Further, the "dyadics" article needs some wikification.
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893:(at least for order-2 tensors) is defined in the
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576:Speaking absurd must be punished. --
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843:InedibleHulk
839:100% Support
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816:— Preceding
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770:— Preceding
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505:—Preceding
468:RainerBlome
918:articles (
776:Eratostene
598:Dankatz316
730:Crowsnest
631:polyadics
578:Javalenok
527:Crowsnest
38:this edit
818:unsigned
784:contribs
772:unsigned
548:see here
507:unsigned
483:Shiznick
943:Maschen
916:dyadics
899:Quondum
695:Quondum
644:Quondum
623:Dyadics
556:Maschen
42:history
34:Dyadics
889:, and
883:tensor
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748:linas
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36:with
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