Let $A,B \in M_n(\mathbb{R})$. We say they are equivalent if there are $P,Q$ invertible such that $A=QBP$ (note this is weaker than similarity). Every matrix is equivalent to a diagonal matrix using row and column operations.
My question is - when is $A$ equivalent to a diagonal matrix via orthogonal matrices? That is, for which $A$ there are a diagonal $D$ and orthogonal $P,Q$ such that $A=PDQ$?
$A$ is orthogonally diagonizable iff it is self adjoint, but this seems to be strictly stronger (it means there $A=P^{T}DP$ for an orthogonal $P$).
