What can we say for two orthogonal matrices to commute?

Question

Suppose that $Q$ is a block diagonal matrix $\mathrm{diag}(Q_1,\dots,Q_r)$ where $Q_i$ is an orthogonal matrix for $i=1,\dots,r$. Let $V$ be an orthogonal matrix such that $V^TQV=Q$. Can we say something about blocks $Q_i$?

Answer 1

Let $V_{i,j}$ denote the block-entries of $V$, where $V$ is partitioned into an $r \times r$ block matrix in such a way that $V_{ii}$ has the same size as $Q_i$ for $i=1,\dots,r$. Then we can write $$ V^TQV = Q \iff QV = VQ \iff Q_i V_{ij} = V_{ij}Q_j, \; i=1,\dots,r. $$ This means that the blocks $Q_i$ are such that $Q_jV_{ij} = Q_i V_{ij}$ for all $1 \leq i,j \leq r$.

Note: If $Q_i$ and $Q_j$ have distinct sizes, then $V_{ij}$ and $V_{ji}$ are not square.

Answer 2

Maybe the thing is to diagonalize $Q$ by a block diagonal unitary $ U $ which is direct sum of $U_i$ with $U_i^*Q_iU_i$ is diagonal for all $i=1,\ldots,r$. Now $U^*VU$ commutes with a diagonal matrix which is a known argument.