Is the product of two orthogonally diagonalizable matrices still orthogonally diagonalizable?

Question

Let $S$ and $T$ be two orthogonally diagonalizable matrices with $ST=TS$. Then is $ST$ still orthogonally diagonalizable? Thank you in advance

Answer 1

No. A real matrix is orthogonally diagonalisable if and only if it is symmetric. If $S$ and $T$ are symmetric, $ST$ is symmetric if and only if $ST=TS$. So, any pair of symmetric matrices $S$ and $T$ such that $ST\ne TS$ can serve as a counterexample.

Answer 2

$A\in{ \mathbb{M}}_n(\mathbb{C})$ orthogonaly diagonalisable by a unitary $U$ is equivalent to $A$ is normal. The classic fact is that the product of two normal matrices is not normal, here since $ST=TS$ with $S$ and $T$ normal they are simultaneously diagonalisable with a unitary matrix $U$ , $TS$ is normal.