If $A = (a_{ij})$ , and $B = (b_{ij})$ , are real $m\!\times\!n$ matrices, their Frobenius product is defined as $$\langle A,\,B \rangle_F \;:=\; \sum_{i,\,j}a_{ij}b_{ij}.$$ It is easily seen that $\langle A,\,B \rangle_F$ , is equal to the trace of the matrix $A^\intercal B$ and $AB^\intercal $ and that the Frobenius product is an inner product of the vector space formed by the $m\!\times\!n$ matrices; it induces the Frobenius norm of this vector space.