A resolution of Paz's conjecture in the presence of a nonderogatory matrix

被引：20

作者：

Guterman, Alexander ^{[1
]}

Laffey, Thomas ^{[2
]}

Markova, Olga ^{[1
]}

Smigoc, Helena ^{[2
]}

机构：

[1] Moscow MV Lomonosov State Univ, Dept Algebra, Fac Mech & Math, Moscow 119991, Russia

[2] Univ Coll Dublin, Sch Math & Stat, Dublin, Ireland

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2018年 / 543卷

基金：

俄罗斯科学基金会;

关键词：

Finite-dimensional algebras; Lengths of sets and algebras; Paz's conjecture; Nonderogatory matrices; COMPLEX MATRICES; IRREDUCIBLE PAIRS; LENGTHS; THEOREM; ALGEBRA; SIZE;

D O I：

10.1016/j.laa.2018.01.002

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let M-n(F) be the algebra of n x n matrices over the field F and let S be a generating set of M-n(F)as an F-algebra. The length of a finite generating set S of M-n(F) is the smallest number k such that words of length not greater than k generate M-n(F) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of M-n(F) cannot exceed 2n - 2. We prove this conjecture under the assumption that the generating set S contains a nonderogatory matrix. In addition, we find linear bounds for the length of generating sets that include a matrix with some conditions on its Jordan canonical form. Finally, we investigate cases when the length 2n - 2 is achieved. (C) 2018 Elsevier Inc. All rights reserved.

引用

页码：234 / 250

页数：17

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