Principal submatrix unitary matrix normal

Question

Given a $n\times n$ unitary matrix $M$ with leading principal submatrix $S$ ($(n-1)\times (n-1)$, codimension $1$), is it true that $S$ is normal?

Answer 1

Try e.g. $$ \pmatrix{1/2 & -1/2 & -\sqrt{2}/2\cr 1/2 & 5/6 & -\sqrt{2}/6\cr \sqrt{2}/2 & -\sqrt{2}/6 & 2/3\cr} $$

Answer 2

It doesn't even have to be selfadjoint. For instance, take $$ M=\frac12\,\begin{bmatrix} 1&-1&1&-1 \\ 1&1&1&1\\ -1&1&1&-1\\ -1&-1&1&1 \end{bmatrix} $$