The Slot Length of a Family of Matrices

Longstaff, W. E.

doi:10.1017/s0004972720000726

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“…With A and F as above, the 'minimum spanning length of A', abbreviated 'msl(A)', was introduced in [6]. (The related notion of 'slot length of A' was introduced in [8].) For every integer k ≥ 1, we let V ′ k (A) denote the linear span of the nonempty words in A of length at most k. If the nonempty words in A span M n (F), the minimum spanning length of A is the least positive integer K such that V ′ K (A) = M n (F).…”

Section: Introductionmentioning

confidence: 99%

-Minimum Spanning Lengths and an Extension to Burnside’s Theorem on Irreducibility

Longstaff¹

2020

Bull. Aust. Math. Soc.

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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.

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Section: Introductionmentioning

confidence: 99%