Hostname: page-component-6b989bf9dc-476zt Total loading time: 0 Render date: 2024-04-14T17:09:39.735Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE LENGTHS OF PAIRS OF COMPLEX MATRICES OF SIZE SIX

Part of: Basic linear algebra

Published online by Cambridge University Press:  08 June 2009

M. S. LAMBROU  and
W. E. LONGSTAFF
M. S. LAMBROU
Affiliation:
Department of Mathematics, University of Crete, Iraklion, Crete, Greece (email: lambrou@math.uoc.gr)
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)
*
✉For correspondence; e-mail: longstaf@maths.uwa.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The length of every pair {A,B} of 6×6 complex matrices is shown to be at most 10, that is, the words in A,B of length at most 10, including the empty word, span the unital algebra generated by A,B. This supports the conjecture that the length of every pair of n×n complex matrices is at most 2n−2, known to be true for n<6.

Keywords

lengthwords

MSC classification

Secondary: 15A30: Algebraic systems of matrices
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 177 - 201
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1] Constantine, D. and Darnall, M., ‘Lengths of finite dimensional representations of PBW algebras’, Linear Algebra Appl. 395 (2005), 175–181.CrossRefGoogle Scholar
[2] Freedman, A., Gupta, R. and Guralnick, R., ‘Shirshov’s theorem and representations of semigroups’, Pacific J. Math., Special Issue (1997), 159–176.CrossRefGoogle Scholar
[3] Longstaff, W. E., ‘Burnside’s theorem: irreducible pairs of transformations’, Linear Algebra Appl. 382 (2004), 247–269.CrossRefGoogle Scholar
[4] Longstaff, W. E., ‘On words in a,b with more than 36 subwords’, unpublished manuscript (2005).Google Scholar
[5] Longstaff, W. E., Niemeyer, A. and Panaia, O., ‘On the lengths of pairs of complex matrices of size at most 5’, Bull. Aust. Math. Soc. 73 (2006), 461–472.CrossRefGoogle Scholar
[6] Paz, A., ‘An application of the Cayley–Hamilton theorem to matrix polynomials in several variables’, Linear Multilinear Algebra 15 (1984), 161–170.CrossRefGoogle Scholar