Talk:Outer product

Associavity
Associativity only in the sense
 * (sa) x b = s(a x b),

with scalar s! Patrick 13:31 Nov 20, 2002 (UTC)

Outer product vs exterior product
Isn't this page redundant? It should be merged with exterior power. This page purports to define wedge product, however wedge product currently links to the exterior algebra page. -- Fropuff 19:34, 2004 May 20 (UTC)


 * The exterior product or wedge product, in the context of exterior algebra, is defined very differently from the outer product presented here today (it might have been different in 2004, qhen you wrote your comment). An outer product returns a matrix. An exterior product creates a multivector. Paolo.dL 13:07, 19 May 2007 (UTC)

Ambiguity (outer, exterior, or tensor?)
This should really be a disambiguation page, Exterior algebra is one possibility, but Tensor product is also valid. RaulMiller 18:52, 3 October 2005 (UTC)


 * I think it is very useful to find the definition of the outer product in a separate page, as well as for dot product and cross product. Perhaps it might be made clearer that, although outer product is a general term, with different meanings depending on the context, most people (as far as I know) tend not to associate it with the exterior product. By the way, I hypothesize that the exterior product was called exterior just to distinguish it from the outer product as defined in this article (see also Talk:Exterior algebra).


 * Notice that, initially, the exterior product was called by Grassmann, the German mathematician who defined it, "ausser produkt" (literally translated as "outer product", where outer has the same meaning as exterior, but is linguistically and phonetically much more similar to the German word ausser). Thus, the expression outer product, initially, was associated with the exterior product rather than with the multiplication defined in this article (the tensor product). Interestingly, as far as I know the opposite is true nowadays. Paolo.dL 13:07, 19 May 2007 (UTC)


 * This is confusing at the moment. In particular, Geometric algebra links here implying that this meaning is the same as the wedge product. I'll add a DAB link to the top of this page and fix that link on Geometric algebra. —Ben FrantzDale (talk) 13:48, 23 January 2009 (UTC)

Wrong note concerning inner product
There is an error here: If W = V, then one can also pair w * (v), which is the inner product. Inner product is a mapping from $$V \times V$$. Here we have mapping $$V^* \times V$$. Natural correspondence between $$V$$ and $$V^*$$ exists only for Hilbert spaces (See Riesz representation theorem for details).

Tenzink (talk) 10:30, 22 April 2008 (UTC)


 * I don't think the correspondence is used here. I think that w* denotes an element of the dual space W*.  (Not the dual of an element of W.)  That said, the section does take a rather idiosyncratic view towards the definition of an inner product.   silly rabbit  (  talk  ) 10:57, 22 April 2008 (UTC)


 * Agree. Tenzink (talk) 10:51, 23 April 2008 (UTC)


 * As regards "Here we have mapping $$V^* \times V$$": May I request that in the future when referring to a mapping, could you please specify both a) the domain and b) the codomain. Thanks. 2601:200:C082:2EA0:782A:5C98:6D2F:A7C5 (talk) 01:47, 21 February 2023 (UTC)

Dyadic Product
Should this page have a link to Dyadic product? Or should the articles be merged? They appear to be the same operation.206.207.225.87 (talk) 15:44, 9 April 2012 (UTC)


 * Dyadic product, Tensor product and Outer product should all be considered as candidates for merging (IMO into Tensor product). — Quondum☏ 10:18, 19 August 2012 (UTC)


 * Possibly. See here. Maschen (talk) 14:18, 21 August 2012 (UTC)

Transform to a disambiguation page?...
Some people have proposed at WP Maths to transform this page into a disambiguation page. I also think it should be, but not immensely fussed if there are objections. Here is my draft, feel free to take or leave, provided a notification is made here and/or WP maths. Maschen (talk) 10:43, 24 August 2012 (UTC)


 * Your draft is still slightly more like a stub than a disambiguation page, and will need minor changes to conform to the guidelines for a disambiguation page. The term is a little unusual in that it seems to be a minority use in every case, and this does not point to any article of the same name, so in a way it is more like a combined dab/redirect. Nevertheless, and I have no objection to it becoming one. — Quondum☏ 20:31, 24 August 2012 (UTC)

Outer-product rank
I think that the explanation regarding the outer product rank is nearly totally incomprehensible. IMHO, a much simpler explanation is that matrix rows represent the same vector scaled differently. If you want to go into a few more details, you can explain that because of the above-mentioned property, for any two rows there obviously always exists a linear combination of these rows that is the zero vector. Likewise, you can nullify any combination of rows by picking two arbitrary rows and nullifying them while assigning zero coefficients for the remaining ones. Finally, you the rank will be one, because if u and v are non-zero, there will be at least one non-zero matrix element. — Preceding unsigned comment added by Srchvrs (talk • contribs) 03:15, 13 July 2016 (UTC)

It's not an incomprehensible explanation. It's slick, which may or may not be a good thing. To get at the real reason rank is 1, "the matrix rows represent the same vector scaled differently" does a better job. But it's heuristic, not a proof, and would take more room if you turned it into a proof. Since the rank of a matrix can be defined in terms of row echelon representation a good compromise might be to state that the row echelon form of an outer product has up to one non-zero row, and why. Colin McRae


 * The reasoning for the rank one property has been changed. The previous argument was missing why the scalar result means rank one. It also failed to mention use of associativity of multiplication being used. Maximal linear independence of columns usually establishes rank, and in the outer product matrix there is only one. — Rgdboer (talk) 22:57, 16 July 2018 (UTC)

Applications
Question about the outer product Applications section. I agree the rank of the outer product matrix is 1. In the applications section it states that the outer product is useful for the Inertia matrix and for vector valued covariance matrices. Something is missing here. In general the inertial matrix is real and symmetric. If the principal axes aligned with the coordinate system, then this is a diagonal matrix with non-zero entries on the diagonal. This matrix is non-singular in general having a rank of 3. How the outer product is related to this matrix isn't clear. Same comment for symmetric covariance matrices that are non-singular. Can anyone clear this up? — Preceding unsigned comment added by 66.62.17.2 (talk) 18:50, 7 June 2018 (UTC)


 * The following was removed:
 * The outer product is useful in computing physical quantities (e.g., the tensor of inertia), and performing transform operations in digital signal processing and digital image processing. It is also useful in statistical analysis for computing the covariance and auto-covariance matrices for two vector-valued random variables.


 * Instead there is now reference to a 2011 text by Steeb & Hardy which has applications of Kronecker products (which include the outer product). — Rgdboer (talk) 22:33, 17 July 2018 (UTC)

The use of outer products for expressing and establishing the completeness of a set of basis vectors could be added here. Applications range from Euclidean space to Hilbert space of square-integrable functions. An intuitive proof of the completeness of a set of Fourier basis functions in terms of the Dirac delta function would be illustrative of how an "arbitrary" function may be represented as a Fourier series.

The section, Spinors, could be vastly expanded to give details as to the application. David in Cincinnati (talk) 14:38, 24 April 2020 (UTC)

Useful information
The article defines "outer product" in terms of coordinates.

It would be useful if the article also included information about whether in some sense the result is independent of coordinates. I hope someone knowledgeable about this subject can add that information.