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

Alexander Guterman; Thomas Laffey; Olga Markova; Helena Šmigoc

doi:10.1016/j.laa.2018.01.002

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

Alexander Guterman, Thomas Laffey, Olga Markova, Helena Šmigoc

Research output: Contribution to journal › Article › peer-review

28 Scopus citations

Abstract

Let M_n(F) be the algebra of n×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.

Original language	English
Pages (from-to)	234-250
Number of pages	17
Journal	Linear Algebra and Its Applications
Volume	543
DOIs	https://doi.org/10.1016/j.laa.2018.01.002
State	Published - 15 Apr 2018
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2018 Elsevier Inc.

Keywords

Finite-dimensional algebras
Lengths of sets and algebras
Nonderogatory matrices
Paz's conjecture

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10.1016/j.laa.2018.01.002

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abstract = "Let Mn(F) be the algebra of n×n matrices over the field F and let S be a generating set of Mn(F) as an F-algebra. The length of a finite generating set S of Mn(F) is the smallest number k such that words of length not greater than k generate Mn(F) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of Mn(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.",

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AU - Guterman, Alexander

AU - Laffey, Thomas

AU - Markova, Olga

AU - Šmigoc, Helena

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

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