How to find the singular values of an orthogonal matrix?

Question

If you have a orthogonal matrix, say $ A \in \mathbb{R}^{nxn}$ how do you find its singular values?

Answer 1

If it is know that the $n\times n$ matrix $A$ obeys $AA^T=I$, then the singular values of $A$ are all equal to $1$. Because we can write an SVD decomposition $A=PDQ$ where $P$ and $Q$ are orthogonal and $D$ diagonal, namely by taking $P=A$, $D=I$, and $Q=I$. Since the identity matrix $I$ is both diagonal and orthogonal, and since $A$ is assumed orthogonal, we know the factorization $A=AII=PDQ$ is a valid singular value decomposition. The singular values of $A$ are thus the diagonal elements of $D=I$, namely $1$.