TITLE: Commutativity of matrix and its transpose
QUESTION [8 upvotes]: If a matrix is symmetric or skew-symmetric it commutes in the obvious way with its transpose.
(For symmetric: $SS^T=S^2$ and $S^TS =S^2$)
The less obvious is the case of commutativity  for orthogonal matrix but such matrix also commutes with its transpose because $RR^T=RR^{-1}= I= R^{-1}R=R^TR$.
Questions:

Are there  other cases when a matrix commutes with its transpose ?   
What is the general property of such matrix which allows it to
commute with its transpose ?

REPLY [9 votes]: Matrices for which $S^*S=SS^*$ holds are called normal. Here $*$ denotes the conjugate transpose. In the case of a real matrix this is equivalent to $SS^T=S^TS$. 
Normal matrices are precisely the ones that are diagonalisable by a unitary transformation, i.e
$$ S = U^* D U $$ for $U$ unitary and $D$ diagonal. 
Unitarity is in general a key property. Even if $S$ is real, an orthogonal transformation will not do in general. Take any non-trivial rotation in the plane for example. It is certainly normal, but has no real eigenvalues.
They can be unitarily diagonalised though.
On the other hand, (skew-) symmetric matrices are an example of real matrices that can be orthogonally diagonalised.
Several equivalent characterisations can be found for example on the Wikipedia page.