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Multilinear maps. Extensions.

  1. Jan 11, 2008 #1


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    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
    music, and about how Frank Sinatra was the last good singer. They also
    talk about Selzer water Melba toast, and that hot new comedian Red Skelton.
    . Maybe this last explains it :smile:)
  2. jcsd
  3. Jan 16, 2008 #2


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    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.
  4. Jan 16, 2008 #3
    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.
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