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THE SLOT LENGTH OF A FAMILY OF MATRICES

Part of: Basic linear algebra

Published online by Cambridge University Press:  05 October 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: [email protected]
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Abstract

We introduce the notion of the slot length of a family of matrices over an arbitrary field ${\mathbb {F}}$ . Using this definition it is shown that, if $n\ge 5$ and A and B are $n\times n$ complex matrices with A unicellular and the pair $\{A,B\}$ irreducible, the slot length s of $\{A,B\}$ satisfies $2\le s\le n-1$ , where both inequalities are sharp, for every n. It is conjectured that the slot length of any irreducible pair of $n\times n$ matrices, where $n\ge 5$ , is at most $n-1$ . The slot length of a family of rank-one complex matrices can be equal to n.

Keywords

irreduciblelinear span

MSC classification

Primary: 15A30: Algebraic systems of matrices
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 260 - 270
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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