One more counterexample on sign patterns

Abstract

The \textit{sepr-sequence} of an $n\times n$ real matrix $A$ is $(s_1,\dots ,s_n)$, where $s_k$ is the subset of those signs of $+,-,0$ that appear in the values of the $k\times k$ principal minors of $A$. The $12\times 12$ matrix $$\left(\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& a_1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& a_2& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& a_3\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& a_4\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& a_5\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& a_6\\ b_1& b_2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ b_3& b_4& 0& 0& b_5& -b_6& 0& 0& 0& 0& 0& 0\\ 0& b_7& b_8& -b_9& b_{10}& b_{11}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& c_1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& c_2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& c_3& 0& 0& 0\end{array}\right)$$ does always have $s_k=\{0,+,-\}$ if $k=3,6,9$ and $s_k=\{0\}$ otherwise, provided that the variables are positive. However, every principal $9\times 9$ minor that is not identically zero can take values of both signs.

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