Vol. 13, No. 6, 2019

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An improved bound for the lengths of matrix algebras

Yaroslav Shitov

Vol. 13 (2019), No. 6, 1501–1507
DOI: 10.2140/ant.2019.13.1501
Abstract

Let S be a set of n × n matrices over a field F. We show that the F-linear span of the words in S of length at most

2nlog2n + 4n

is the full F-algebra generated by S. This improves on the n2 3 + 2 3 bound by Paz (1984) and an O(n32) bound of Pappacena (1997).

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Keywords
matrix theory, finite-dimensional algebras, generating sets
Mathematical Subject Classification 2010
Primary: 15A03
Secondary: 15A30
Milestones
Received: 14 November 2018
Revised: 6 March 2019
Accepted: 14 May 2019
Published: 18 August 2019
Authors
Yaroslav Shitov
Moscow
Russia