One more counterexample on sign patterns

TL;DR

This paper presents a specific 12x12 matrix example demonstrating particular sign patterns in principal minors, highlighting nuanced behaviors in sign patterns of matrices with positive variables.

Contribution

It provides a counterexample matrix showing fixed sign patterns for certain minors while others can vary in sign, advancing understanding of sign pattern behaviors.

Findings

The matrix always has sign pattern {0,+,-} for minors of size 3, 6, 9.

Other minors of size 3, 6, 9 can take both positive and negative values.

Minors of other sizes are always zero.

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.

Equations

$$\left(\begin{array}[]{cccccc|ccc|ccc}0&0&0&0&0&0&0&0&0&a_{1}&0&0\\ 0&0&0&0&0&0&0&0&0&0&a_{2}&0\\ 0&0&0&0&0&0&0&0&0&0&0&a_{3}\\ 0&0&0&0&0&0&0&0&0&0&0&a_{4}\\ 0&0&0&0&0&0&0&0&0&0&0&a_{5}\\ 0&0&0&0&0&0&0&0&0&0&0&a_{6}\\ \hline\cr b_{1}&b_{2}&0&0&0&0&0&0&0&0&0&0\\ b_{3}&b_{4}&0&0&b_{5}&-b_{6}&0&0&0&0&0&0\\ 0&b_{7}&b_{8}&-b_{9}&b_{10}&b_{11}&0&0&0&0&0&0\\ \hline\cr 0&0&0&0&0&0&c_{1}&0&0&0&0&0\\ 0&0&0&0&0&0&0&c_{2}&0&0&0&0\\ 0&0&0&0&0&0&0&0&c_{3}&0&0&0\end{array}\right)$$

Full text

[TABLE]

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.

This is a counterexample to Conjecture 3.1 in [1]. The claims can be checked with what I hope is a user friendly Wolfram Mathematica worksheet [2].