Can Multilinear Maps Be Represented with Matrices?

[WWGD]

Hi, everyone:

There are standard ways of representing linear and bilinear maps.
between (fin. dim) vector spaces, after choosing a basis .Linear maps
are represented by columns T(vi) , for a basis {v1,...,vn} (assume B
defined on VxV ), bilinear maps B(x,y) with the matrix Bij=(B(ei,ej))
Is there a way of representing 3-linear, 4-linear, etc. maps with
matrices?. I have played around with matrices T(ei,ej,ek), but
I cannot see how to get a real number as a product of 3 matrices.
Any ideas?.

P.S: I don't know how to setup the spacing.In this forum I was asked
to not leave spacing. In other forums, people complain when I don't
leave spacing, because the lack of spaces force them to strain their
eyes ( where they also complain about how kids today don't understand
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. Maybe this last explains it )

 

[WWGD]

Here is the answer, given to me by someone else (Prof. R. Israel), in case anyone
else is interested:

Suppose T is an n-linear map from V^n to W, and B = {b_i: i=1..m} is a basis
of V. Then the m^n vectors T(b_i) = T(b_{i_1},...,b_{i_n}) for n-tuples
i = (i_1,...,i_n) in {1..m}^n determine T, since if each
x_j = sum_{k = 1}^m c_{j,k} b_k,
T(x_1,...,x_n) = sum_{i in {1..m}^n} product_{j=1}^n c_{j, i_j} T(b_i).

A bilinear map from V^2 to the reals R, for example, can be represented by an
m x m matrix of real numbers: each entry has a pair of indices. A trilinear
map from V^3 to the reals would be represented by a triply-indexed array
of real numbers, rather than a matrix.

 

[mathboy]

A p-linear map can be represented by a matrix (tensor) with p indices. Most of the basic theorems with linear maps and bilinear maps are generalized to the p-linear case. The proofs remain the same too, but reading and writing the proofs is really messy because of all the indices.