Let $X, Y ∈ R^{n × m}$. We are looking for an orthogonal matrix $R ∈ M_n (R)$ that minimizes $\|RX - Y\|^2=\operatorname{tr}((RX - Y)^T (RX - Y))$.

Question

I'm trying to show that the wanted matrix is $R = VU^T$, where $XY^T = UΣV^T$ is the singular value decomposition, but I'm having trouble understanding what it means or how to start.