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Math Help - proof of inner product vector spaces

  1. #1
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    proof of inner product vector spaces

    Let V be the vector space of m x n matrices over R, Prove that

    f(A,B)=trace(BtA) defines an inner product in V

    ive manged to satisfy one of the three terms which was f(u,v)=f(v,u)
    just cant seem to prove the others

    (Bt is b transpose)
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  2. #2
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    Quote Originally Posted by feage7 View Post
    Let V be the vector space of m x n matrices over R, Prove that

    f(A,B)=trace(BtA) defines an inner product in V

    ive manged to satisfy one of the three terms which was f(u,v)=f(v,u)
    just cant seem to prove the others

    (Bt is b transpose)
    1. The diagonal elments of the product B'A are the inner products of the col of B with the cols of A, so for A'A the diagonal elements are the inner products of the col of A, and so >0, hence tr(A'A)>=0, so f(A,A)>=0, and f(A,A)=0 only iff every col of A is zero, or that A is the mxn zero matrix.

    2. (B+C)'A = B'A + C'A, and trace is additive so:

    f(A,B+C)= tr((B+C)'A) = tr(B'A) + tr(C'A) = f(A,B) + f(A,C)

    3. for all alpha in R, f(alpha*A,B) = tr(B'(alpha*A)) = tr(alpha* B'A) = alpha* tr(B'A),
    so: f(alpha*A,B) = alpha*f(A,B).

    RonL
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