FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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 $GMr(A)$ be the row Gondran--Minoux rank of a matrix, $GMc(A)$ be the column Gondran--Minoux rank, and $d(A)$ 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 $(m$´ n)-matrix $B$ with different row and column Gondran--Minoux ranks. We prove that in the case $GMr(B)\; >\; GMc(B)$ the minimal $m$ and $n$ are equal to $5$ and $6$, respectively, and in the case $GMc(B)\; >\; GMr(B)$ the numbers $m\; =\; 6$ and $n\; =\; 5$ are minimal. An example of a matrix $A$Î M_{5 ´ 6}(R_max) such that $GMr(A)\; =\; GMc(At)\; =\; 5$ and $GMc(A)\; =\; GMr(At)\; =\; 4$ is provided. It is proved that $p\; =\; 5$ and $q\; =\; 6$ are the minimal numbers such that there exists an $(p$´ q)-matrix with different row Gondran--Minoux and determinantal ranks.

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