A sign pattern with non-zero elements on the diagonal whose minimal rank realizations are not diagonalizable over the complex numbers

Abstract

The rank of the $9\times 9$ matrix $$ \left( \begin{array}{cccc|c|cccc} 1&1&0&0&1&0&0&0&0\\ 1&1&0&0&0&0&0&0&0\\ 0&0&1&1&1&0&0&0&0\\ 0&0&1&1&0&0&0&0&0\\\hline 0&0&0&0&1&0&1&0&1\\\hline 0&0&0&0&0&1&1&0&0\\ 0&0&0&0&0&1&1&0&0\\ 0&0&0&0&0&0&0&1&1\\ 0&0&0&0&0&0&0&1&1 \end{array} \right) $$ is $6$. If we replace the ones by arbitrary non-zero numbers, we get a matrix $B$ with $\operatorname{rank} B\geqslant6$, and if $\operatorname{rank} B=6$, the $6\times 6$ principal minors of $B$ vanish.