# One more counterexample on sign patterns

Research paper by **Yaroslav Shitov**

Indexed on: **03 Feb '19**Published on: **03 Feb '19**Published in: **arXiv - Mathematics - Combinatorics**

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#### Abstract

The \textit{sepr-sequence} of an $n\times n$ real matrix $A$ is
$(s_1,\ldots,s_n)$, 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 $$\left(\begin{array}{cccccc|ccc|ccc} 0&0&0&0&0&0&0&0&0&a_1&0&0\\
0&0&0&0&0&0&0&0&0&0&a_2&0\\ 0&0&0&0&0&0&0&0&0&0&0&a_3\\
0&0&0&0&0&0&0&0&0&0&0&a_4\\ 0&0&0&0&0&0&0&0&0&0&0&a_5\\
0&0&0&0&0&0&0&0&0&0&0&a_6\\\hline b_1&b_2&0&0&0&0&0&0&0&0&0&0\\
b_3&b_4&0&0&b_5&-b_6&0&0&0&0&0&0\\
0&b_7&b_8&-b_9&b_{10}&b_{11}&0&0&0&0&0&0\\\hline 0&0&0&0&0&0&c_1&0&0&0&0&0\\
0&0&0&0&0&0&0&c_2&0&0&0&0\\ 0&0&0&0&0&0&0&0&c_3&0&0&0 \end{array}\right)$$ 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.