Does a Householder Matrix commute with a unitary matrix?

Question

If $A$ is a unitary matrix and $P=I-2\alpha\alpha^H(0\neq\alpha\in\mathbb F^{n\times 1},\lVert\alpha\rVert=1)$, then does $PA$ equals to $AP$?

Answer 1

We can see that equality would require $(Aa)(Aa)^H=aa^H$ for any unitary $A$ and unit $a$, which is clearly false for, say, $a=(1, 0)^T$ and $$A=\begin{bmatrix}\cos\frac\pi 4 & \sin \frac\pi 4 \\ -\sin \frac\pi 4 & \cos\frac\pi 4\end{bmatrix}$$

Indeed, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin (Wikipedia), so, even in 2D, it's quite intuitive that they don't necessarily commute with a generic unitary matrix which may represent a rotation.