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IRREDUCIBLE FAMILIES OF COMPLEX MATRICES CONTAINING A RANK-ONE MATRIX

Part of: Basic linear algebra

Published online by Cambridge University Press:  16 January 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 email [email protected]
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Abstract

We show that an irreducible family ${\mathcal{S}}$ of complex $n\times n$ matrices satisfies Paz’s conjecture if it contains a rank-one matrix. We next investigate properties of families of rank-one matrices. If ${\mathcal{R}}$ is a linearly independent, irreducible family of rank-one matrices then (i) ${\mathcal{R}}$ has length at most $n$, (ii) if all pairwise products are nonzero, ${\mathcal{R}}$ has length 1 or 2, (iii) if ${\mathcal{R}}$ consists of elementary matrices, its minimum spanning length $M$ is the smallest integer $M$ such that every elementary matrix belongs to the set of words in ${\mathcal{R}}$ of length at most $M$. Finally, for any integer $k$ dividing $n-1$, there is an irreducible family of elementary matrices with length $k+1$.

Keywords

irreduciblerank oneelementary matrices

MSC classification

Primary: 15A30: Algebraic systems of matrices
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 226 - 236
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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