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Matrices with different Gondran-Minoux and determinantal ranks over max-algebras
Journal of Mathematical Sciences. 2009. Vol. 163. No. 5. P. 598–624.
Shitov Y.
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∈M5×6(Rmax) 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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