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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
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
MSC classification
Primary:
15A30: Algebraic systems of matrices
Information
- Type
- Research Article
- Information
- Copyright
- © 2020 Australian Mathematical Publishing Association Inc.
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