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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the lengths of irreducible pairs of complex matrices
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by W. E. Longstaff and Peter Rosenthal PDF
Proc. Amer. Math. Soc. 139 (2011), 3769-3777 Request permission


The length of a pair of matrices is the smallest integer $l$ such that words in the matrices with at most $l$ factors span the unital algebra generated by the pair. Upper bounds for lengths have been much studied. If $B$ is a rank one $n\times n$ (complex) matrix, the length of the irreducible pair $\{A,B\}$ is $2n-2$ and the subwords of $A^{n-1}BA^{n-2}$ form a basis for $M_n(\mathbb {C})$. New examples are given of irreducible pairs of $n\times n$ matrices of length $n$. There exists an irreducible pair of $5\times 5$ matrices of length $4$. We begin the study of determining lower bounds for lengths.
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Additional Information
  • W. E. Longstaff
  • Affiliation: School of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
  • Email:
  • Peter Rosenthal
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
  • Email:
  • Received by editor(s): March 1, 2010
  • Published electronically: June 13, 2011
  • Communicated by: Marius Junge
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 3769-3777
  • MSC (2010): Primary 15A30; Secondary 47L05
  • DOI:
  • MathSciNet review: 2823023