Take given matrices $A, B \in \mathbb{R}^{m \times n}$. We want to find an orthogonal matrix $Q$ (in $\mathbb{R}^{n \times n}$) to maximize the inner product $\langle A, QB \rangle$.
So I know that if this problem were instead to find $Q, R$ to maximize $\langle A, QBR \rangle$, we would just take $Q = U_A U_B^T$ and $R = V_B V_A^T$ from the SVDs of $A, B$. That actually achieves the maximum $\textrm{Tr}(\Sigma_A \Sigma_B)$.
But I'm not sure how to proceed when $B$ is only multiplied from the left. I suspect that we would still take $Q = U_A U_B^T$ so the inner product is $\textrm{Tr}(\Sigma_A \Sigma_B V_B^T V_A)$, but I have no idea how to show that this is indeed the maximum. My thought is that $V_B^T V_A$ will always be in the inner product, but why?
Would really appreciate answers centered around the SVD, as well as general intutition here. Thank you!
as mentioned above this is similar to the Procrustes problem, however, it is possible to provide another solution not based on the SVD decomposition, the complete proof can be found in this paper, in Lemma 4.2.2, page 82. I hope you find it useful.