# Thread: proof of inner product vector spaces

1. ## 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)

2. Originally Posted by feage7
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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### prove trace is inner product

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