Given orthogonal matrix $P$, is $P \circ P$ invertible?

Question

Let $P$ be a orthogonal matrix, i.e., $P^T P = P P^T =I.$ Then can we say that $P \circ P$ is invertible?

P.S: $A \circ B$ is the elementwise product of matrices $A$ and $B$.

Answer 1

No.

My gut reaction to reading any question like this is skepticism: elementwise product is an "unnatural" operation on matrices, and rarely bears much relation to ordinary matrix multiplication. So I wouldn't expect the latter to give much guarantees about the former.

So I try to turn my skepticism into a concrete counterexample. Well, $P \circ P$ is a squaring of all of the entries of $P$. An easy example of a singular matrix is one where all of the entries are equal. The zero matrix won't work, obviously, so what about the ones matrix? I just need a basis of vectors that consist of $\pm 1$ and are mutually orthogonal. For example, $(1,1)$ and $(1,-1)$ in $\mathbb{R}^2$.

So now with a little clean-up to correct for norms, I have the counterexample: $$P = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1\\1 & -1\end{bmatrix}.$$