(FUNDAMENTAL AND APPLIED MATHEMATICS)

2008, VOLUME 14, NUMBER 4, PAGES 231-268

## Matrices with different Gondran--Minoux and determinantal ranks over max-algebras

Ya. N. Shitov

Abstract

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Let $GMr\left(A\right)$ be the row Gondran--Minoux rank of a matrix, $GMc\left(A\right)$ be the column Gondran--Minoux rank, and $d\left(A\right)$ be the determinantal rank, respectively. The following problem was posed by M. Akian, S. Gaubert, and A. Guterman: Find the minimal numbers $m$ and $n$ such that there exists an $\left(m$´ n)-matrix $B$ with different row and column Gondran--Minoux ranks. We prove that in the case $GMr\left(B\right) > GMc\left(B\right)$ the minimal $m$ and $n$ are equal to $5$ and $6$, respectively, and in the case $GMc\left(B\right) > GMr\left(B\right)$ the numbers $m = 6$ and $n = 5$ are minimal. An example of a matrix $A$Î M5 ´ 6(Rmax) such that $GMr\left(A\right) = GMc\left(At\right) = 5$ and $GMc\left(A\right) = GMr\left(At\right) = 4$ is provided. It is proved that $p = 5$ and $q = 6$ are the minimal numbers such that there exists an $\left(p$´ q)-matrix with different row Gondran--Minoux and determinantal ranks.

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