2016/03/04 by Marko Kandić, Kandić, Marko, Klemen Šivic +1 · 1 citation
Mathematics · Engineering · Computer Science · #Advanced Topics in Algebra #graph theory and CDMA systems #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1603.08413
Gerstenhaber's theorem states that the dimension of the unital algebra\ngenerated by two commuting n\× n matrices is at most n. We study the\nanalog of this question for positive matrices with a positive commutator. We\nshow that the dimension of the unital algebra generated by the matrices is at\nmost \(n(n+1))/(2) and that this bound can be attained. We also consider\nthe corresponding question if one of the matrices is a permutation or a\ncompanion matrix or both of them are idempotents. In these cases, the upper\nbound for the dimension can be reduced significantly. In particular, the unital\nalgebra generated by two semi-commuting positive idempotent matrices is at most\n9-dimensional. This upper bound can be attained.\n