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 $$\left(\begin{array}{ccccccccc}1& 1& 0& 0& 1& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 1& 0& 1\\ 0& 0& 0& 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 0& 0& 0& 1& 1\end{array}\right)$$ is $6$. If we replace the ones by arbitrary non-zero numbers, we get a matrix $B$ with $\mathrm{rank}B⩾6$, and if $\mathrm{rank}B=6$, the $6\times 6$ principal minors of $B$ vanish.

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