Is the tensor product of a row vector ${\displaystyle \mathbf {v} }$ with a column vector ${\displaystyle \mathbf {u} }$ still called a dyadic product? For example:

${\displaystyle \mathbf {v} \otimes \mathbf {u} ={\begin{bmatrix}v_{1}&v_{2}&v_{3}\end{bmatrix}}\otimes {\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}={\begin{bmatrix}v_{1}u_{1}&v_{2}u_{1}&v_{3}u_{1}\\v_{1}u_{2}&v_{2}u_{2}&v_{3}u_{2}\\v_{1}u_{3}&v_{2}u_{3}&v_{3}u_{3}\end{bmatrix}}}$

The indices in the definitions would have to be swapped in that case. --RainerBlome 21:28, 27 August 2005 (UTC)

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 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 representing dyadic products, not in defining them. Shiznick 06:02, 8 May 2007 (UTC)

## Proposed merge into Outer product

Why/how is this any different than the Outer Product? Should it be merged? —Preceding unsigned comment added by 132.170.160.64 (talk) 22:56, 31 March 2008 (UTC)

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. Crowsnest (talk) 23:50, 31 March 2008 (UTC)
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 - see here. The operation there (outer product) is no different to this dyadic product. Maschen (talk) 13:38, 21 August 2012 (UTC)
Speaking absurd must be punished. --Javalenok (talk) 08:58, 10 September 2010 (UTC)
An outer product in fact yields a second-order tensor/matrix as well...In all honesty, I see no reason for this separate article. Dankatz316 (talk) 22:30, 15 May 2011 (UTC)
I oppose such a merge. Looking at Dyadics, it would seem that there is a similarity between a dyadic product and the tensor product of two vectors. To call the dyadic product a second-order tensor may not be such a good idea though: dyadics seems to be part of polyadics, an algebraic system similar to but not quite the full tensor algebra. 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 similarity.Quondum 17:17, 15 January 2012 (UTC)
merge -- in what way is it similar to but not quite the full tensor algebra ? 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. linas (talk) 18:55, 3 July 2012 (UTC)
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 tensor algebra" 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. — Quondum 10:13, 19 August 2012 (UTC)

1. "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
2. "dyadic product" is the most used operation of the "dyadics", and deserves an article on its own.

Further, the "dyadics" article needs some wikification. Crowsnest (talk) 17:59, 18 July 2008 (UTC)

support merge -- 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. linas (talk) 18:48, 3 July 2012 (UTC)

## Undefined multiplication in 4th and 5th identities

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 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 — Preceding unsigned comment added by Eratostene (talkcontribs) 17:56, 5 October 2011 (UTC)

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. 68.164.80.215 (talk) 18:19, 2 August 2012 (UTC)

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. — Preceding unsigned comment added by 2.123.253.142 (talk) 23:30, 20 January 2012 (UTC)

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 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 100% Support whatever you think is best for everyone. InedibleHulk (talk) 07:13, 23 August 2012 (UTC)

## "order" vs "rank"?

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? 98.223.186.49 (talk) 09:57, 25 May 2012 (UTC)

The words are not used consistently historically (especially rank being used for two distinct concepts), but usage seems IMO to be converging on the usage in this article (at least within Wikipedia). Preferably, the tensor article should be updated to be clearer on the definition of order, and rank (at least for order-2 tensors) is defined in the article on matrices. The links in the lead here are a mess, though. — Quondum 15:51, 27 May 2012 (UTC)

## For the record...

Given the consensus to merge this article into the other dyadics articles (see WP Maths), the merge is effectivley pre-done and can be inserted into dyadic tensor, although I plan to add/tweak more.

It conflicts the idea of merging this article into outer product, 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 this section. Maschen (talk) 11:25, 23 August 2012 (UTC)