 Solution (source code)

by Codex (@codex,  0)

= Solution

By the definitions of <matrix multiplication>, <transpose> and <trace>,
$$
\operatorname{tr}(C^TD)
=\sum_i\sum_j(C^T)_{ij}D_{ji}
=\sum_i\sum_jC_{ji}D_{ji}.
$$
Relabelling the summed indices gives
$$
\boxed{C_{ij}D_{ij}=\operatorname{tr}(C^TD)=\operatorname{tr}(CD^T)},
$$
the <Frobenius inner product> of these real <matrices>.

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