Solution
= Solution
The <matrix trace> is the sum of the diagonal entries:
$$
\operatorname{tr}A=\sum_{i=1}^n A_{ii}.
$$
Using the definitions of <matrix multiplication> and <matrix transpose>,
$$
\operatorname{tr}(A^TB)
=\sum_{i=1}^n\sum_{j=1}^n A_{ji}B_{ji}
=\sum_{i=1}^n\sum_{j=1}^n B_{ji}A_{ji}
=\boxed{\operatorname{tr}(B^TA)}.
$$
This is the symmetry of the real <Frobenius inner product>.