Centralizer of $SO(n)$ Given the set $M(n,\mathbb C)$ of all complex $n\times n$ matrices, what's the centralizer of $SO(n)$ in $M(n,\mathbb C)$?
For $n=2$, the centralizer must be the matrices $A$ such that $RA=AR$ where $R$ is a rotation matrix. Since 2D rotations commute, I can see $A$ is probably a rotation matrix itself.
But let's say $n \ge 3$. Then rotations don't commute in general. Can you still find a nontrivial matrix $A$ that commutes with $R$, or is $A$ just a multiple of the identity?

If $A$ centralises $SO(n)$, it commutes with every $R$ of the form $R=P\left[\pmatrix{0&-1\\ 1&0}\oplus I_{n-2}\right]P^T$ where $P$ is a permutation matrix. Therefore, when $n\ge3$, $A$ must be a diagonal matrix. Consider the equality $AR=RA$ again, we can further infer that $A$ is a scalar multiple of $I_n$.
