Non-commuting PSD matrices with fixed diagonals

Question

Consider the set of $n\times n$ symmetric positive semi-definite matrices with all-ones on the diagonals. Is it possible to find two such matrices that do not commute? This seems impossible in the $n=2$ case.

Answer 1

It's not possible in the $2 \times 2$ case because all $2 \times 2$ symmetric PSD matrices with all $1$'s on the diagonal are of the form $$\begin{bmatrix}1 & a \\ a & 1\end{bmatrix}$$ for some $a \in [-1,1]$, and these all have the same eigenvectors, namely $\begin{bmatrix}1 \\ 1 \end{bmatrix}$ and $\begin{bmatrix}1 \\ -1 \end{bmatrix}$. Hence, they all commute.

For the $3 \times 3$ case, it is easy to find counterexamples. One such counterexample is $$\begin{bmatrix}1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \quad \text{and} \quad \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1\end{bmatrix}.$$