Let $R \in O(N)$ be an orthogonal map in $\mathbb{R}^N$. Write $N = N_1 + N_2$. I am wondering if is it possible to find a map $T \in O(N_1)$ and $L \in O(N_2)$ such that $R(x,y) = (T(x), L(y))$. If it is possible, how to prove it? If this is not true, is it easy to find a counterexample?
2026-03-28 09:32:56.1774690376
Decompose a orthogonal map as orthogonal maps in two directions
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