Talk:Multilinear map

TooMuchMath's comments

It would be nice to define a multilinear map as a map of R-modules to an R-module that is linear in each variable as I believe this is the most general sense of the term, however I don't know of any examples of such maps where the codomain is not R itself.

Also it may be useful to define symmetric, antisymmetric and alternating multilinear functions. I believe the appropriate definitions are

A multilinear function is called symmetric if D(a_1,...,a_i,...,a_j,...,a_n) = D(a_1,...,a_j,...,a_i,...,a_n)

A multilinear function is called antisymmetric if D(a_1,...,a_i,...,a_j,...,a_n) = -D(a_1,...,a_j,...,a_i,...,a_n)

A multilinear function is called alternating if D(a_1,...,a_i,...,a_i,...,a_n) = 0

I would like someone else to verify this before I put it up. Moreover, the relationships between these seem to be that alternating implies antisymmetric (regardless of the characteristic of the field) and antisymmetric implies alternating if the characteristic is not 2. Also, symmetric = antisymmetric iff the characteristic is 2.

TooMuchMath 01:33, 29 January 2006 (UTC)[reply]

Yes. [1] Black Carrot (talk) 16:44, 13 February 2008 (UTC)[reply]

Question

A multilinear map is a function of several vector variables to what? The example given in the article, the inner product function, takes two vectors and returns a complex or real number. Would it still be a multilinear map if it returned a vector, or indeed a pair of vectors? —Egriffin (talk) 21:15, 23 January 2008 (UTC)[reply]

Right, vector valued maps --kiddo (talk) 03:38, 25 January 2008 (UTC).[reply]

In fact check it out: Vector-valued differential form--kiddo (talk) 03:40, 25 January 2008 (UTC)[reply]

Definition of antisymmetric multlinear map

The article uses antisymmetric without a definition. The first time it is used, it links to Wikipedia's Linear Algebra page, which is very bad because that page never mentions the word "antisymmetric" or "anti-symmetric" at all. The second time it's used, it links to Skew-symmetric matrix, which is not what we mean here (but may be a special case, I don't know).

What is actually meant is explained in this Math StackExchange answer -- the idea is to distinguish alternating, where it is zero when two columns are equal, from antisymmetric, where switching two columns negates the answer. But I don't think this is defined on Wikipedia. The page for alternating multlinear map (here has the same problem, referring to anti-symmetric maps without defining them.

So we should define this somewhere. Could we define it in this article? Cstanford.math (talk) 19:26, 2 March 2017 (UTC)[reply]

"Multilinear" listed at Redirects for discussion

A discussion is taking place to address the redirect Multilinear. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 June 24#Multilinear until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 (talk) 18:30, 24 June 2020 (UTC)[reply]