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$\textit{h}$-MINIMUM SPANNING LENGTHS AND AN EXTENSION TO BURNSIDE’S THEOREM ON IRREDUCIBILITY

Part of: Basic linear algebra

Published online by Cambridge University Press:  02 December 2020

W. E. LONGSTAFF Open the ORCID record for W. E. LONGSTAFF [Opens in a new window]
W. E. LONGSTAFF*
Affiliation:
11 Tussock Crescent, Elanora, Queensland4221, Australia e-mail: bill.longstaff@alumni.utoronto.ca
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Abstract

We introduce the $\textbf{h}$ -minimum spanning length of a family ${\mathcal A}$ of $n\times n$ matrices over a field $\mathbb F$ , where $\textbf{h}$ is a p-tuple of positive integers, each no more than n. For an algebraically closed field $\mathbb F$ , Burnside’s theorem on irreducibility is essentially that the $(n,n,\ldots ,n)$ -minimum spanning length of ${\mathcal A}$ exists if ${\mathcal A}$ is irreducible. We show that the $\textbf{h}$ -minimum spanning length of ${\mathcal A}$ exists for every $\textbf{h}=(h_1,h_2,\ldots , h_p)$ if ${\mathcal A}$ is an irreducible family of invertible matrices with at least three elements. The $(1,1, \ldots ,1)$ -minimum spanning length is at most $4n\log _{2} 2n+8n-3$ . Several examples are given, including one giving a complete calculation of the $(p,q)$ -minimum spanning length of the ordered pair $(J^*,J)$ , where J is the Jordan matrix.

Keywords

irreduciblelinear span

MSC classification

Primary: 15A30: Algebraic systems of matrices
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 78 - 93
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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