The slot length of a family of matrices

Authors

Keywords:

irreducible, linear span

Abstract

We introduce the notion of the slot length of a family of matrices over an arbitrary field $\DF$. 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 equal $n$.

Author Biography

W. E. Longstaff

Retired Professor( University of Western Australia, 1976-2010)

Published

2021-03-10

Issue

Vol. 103 No. 2 (2021)

Section

Articles
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