Inner product of two matrices

Let M(m,n)[R] be the vector space of m by n matrices with elements in R and scalar field R.

Let A, B, C elements in M(m,n)[R] and x,y elements in R. Define the map

f: V x V -> R by (A,B) |-> Tr(B^T * A).

Since
- (xA+yB,C) = x(A,C) + y(B,C)
- (A,B) = (B,A)
- (A,A) >= 0
(proofs are trivial)

f is an inner product.