Frobenius product of A and B is the trace of A transpose B

For two matrices and , their product is given as

Hence, for , we obtain an matrix where

The trace of this matrix is the sum of its diagonal elements

Plugging this into our formula for and letting , we can see that the trace of is

But now recall that the Frobenius inner product is

Renaming to and to in the the formula for , we see that it and the definition of become identical. Hence we have shown that