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arXiv · 1312.5369

Sign patterns of rational matrices with large rank

Yaroslav Shitov
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Abstract

Let $A$ be a real matrix. The term rank of $A$ is the smallest number $t$ of lines (that is, rows or columns) needed to cover all the nonzero entries of $A$. We prove a conjecture of Li et al. stating that, if the rank of $A$ exceeds $t-3$, there is a rational matrix with the same sign pattern and rank as those of $A$. We point out a connection of the problem discussed with the Kapranov rank function of tropical matrices, and we show that the statement fails to hold in general if the rank of $A$ does not exceed $t-3$.

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Yaroslav Shitov. 2013-12-18. Sign patterns of rational matrices with large rank. https://arxiv.org/abs/1312.5369

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