Finding all orthogonal matrices commuting with a positive-definite matrix

Question

Given $M$ a symmetric positive-definite matrix, I'd like to characterise the orthogonal matrices $Q$ commuting with $M$: $MQ=QM$. $Q$ and $M$ commute if and only if they are simultaneously diagonalisable ($M$ is diagonalisable in $\mathbb{R}$ since it is symmetric) but I wonder if it possible to go a bit further in the characterisation.

Answer 1

It is not true that the matrices must be simultaneously diagonalizable. For example, if $M = I$ then all orthogonal matrices commute with $M$ but not all orthogonal matrices are diagonalizable over $\mathbb{R}$. If the distinct eigenvalues of $M$ are $\lambda_1, \dots, \lambda_k$ with multiplicity $n_1, \dots, n_k$, then with respect to an orthonormal basis of eigenvectors of $M$, the map $\vec{x} \mapsto Q\vec{x}$ will be represented by a block diagonal matrix with blocks $Q_i$ of size $n_i$ and each block $Q_i$ will be orthogonal. In other words, $Q$ will be similar to $\operatorname{diag}(Q_1, \dots, Q_k)$. I'm not sure if you can get a better description.