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On the lengths of irreducible pairs of complex matrices


Authors: W. E. Longstaff and Peter Rosenthal
Journal: Proc. Amer. Math. Soc. 139 (2011), 3769-3777
MSC (2010): Primary 15A30; Secondary 47L05
DOI: https://doi.org/10.1090/S0002-9939-2011-11149-3
Published electronically: June 13, 2011
MathSciNet review: 2823023
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Abstract: 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: longstaf@maths.uwa.edu.au

Peter Rosenthal
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
Email: rosent@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11149-3
Keywords: Length, words
Received by editor(s): March 1, 2010
Published electronically: June 13, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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