Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-22T11:40:36.052Z Has data issue: false hasContentIssue false
  • English
  • Français

On the lengths of pairs of complex matrices of size at most five

Published online by Cambridge University Press:  17 April 2009

W. E. Longstaff ,
A. C. Niemeyer  and
Oreste Panaia
W. E. Longstaff
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia, e-mail: longstaf@maths.uwa.edu.au, alice@maths.uwa.edu.au, oreste@maths.uwa.edu.au
A. C. Niemeyer
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia, e-mail: longstaf@maths.uwa.edu.au, alice@maths.uwa.edu.au, oreste@maths.uwa.edu.au
Oreste Panaia
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia, e-mail: longstaf@maths.uwa.edu.au, alice@maths.uwa.edu.au, oreste@maths.uwa.edu.au
Rights & Permissions [Opens in a new window]

Extract

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 n×n complex matrices is at most 2n − 2, if n ≤ 5. That is, for n ≤ 5, the (possibly empty) words in A, B of length at most 2n − 2 span the unital algebra  generated by A, B. For every positive integer m there exist m×m complex matrices C, D such that the length of the pair {C, D} is 2m – 2.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 3 , June 2006 , pp. 461 - 472
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Freedman, A., Gupta, R. and Guralnick, R., ‘Shirshov's theorem and representations of semigroups’, Pacific J. Math., Special Issue (1997), 159–176.CrossRefGoogle Scholar
[2]Longstaff, W.E., ‘Burnside's Theorem: irreducible pairs of transformations’, Linear Algebra Appl. 382 (2004), 247–269.CrossRefGoogle Scholar
[3]Pappacena, C.J., ‘An upper bound for the length of a finite-dimensional algebra’, J. Algebra 197 (1997), 535–545.CrossRefGoogle Scholar
[4]Paz, A., ‘An application of the Cayley-Hamilton theorem to matrix polynomials in several variables’, Linear and Multilinear Algebra 15 (1984), 161–170.CrossRefGoogle Scholar
[5]Procesi, C., Rings with polynomial identities (Marcel Dekker, New York, 1973).Google Scholar