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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- L. Brickman and P. A. Fillmore,
*The invariant subspace lattice of a linear transformation*, Canadian J. Math.**19**(1967), 810–822. MR**213378**, DOI https://doi.org/10.4153/CJM-1967-075-4 - D. Constantine and M. Darnall,
*Lengths of finite dimensional representations of PBW algebras*, Linear Algebra Appl.**395**(2005), 175–181. MR**2112883**, DOI https://doi.org/10.1016/j.laa.2004.08.006 - A. Freedman, R. N. Gupta, and R. M. Guralnick,
*Shirshov’s theorem and representations of semigroups*, Pacific J. Math.**Special Issue**(1997), 159–176. Olga Taussky-Todd: in memoriam. MR**1610851**, DOI https://doi.org/10.2140/pjm.1997.181.159 - M. S. Lambrou and W. E. Longstaff,
*On the lengths of pairs of complex matrices of size six*, Bull. Aust. Math. Soc.**80**(2009), no. 2, 177–201. MR**2540352**, DOI https://doi.org/10.1017/S0004972709000112 - W. E. Longstaff,
*Burnside’s theorem: irreducible pairs of transformations*, Linear Algebra Appl.**382**(2004), 247–269. MR**2050111**, DOI https://doi.org/10.1016/j.laa.2003.12.043 - W. E. Longstaff, A. C. Niemeyer, and Oreste Panaia,
*On the lengths of pairs of complex matrices of size at most five*, Bull. Austral. Math. Soc.**73**(2006), no. 3, 461–472. MR**2230653**, DOI https://doi.org/10.1017/S0004972700035462 - Christopher J. Pappacena,
*An upper bound for the length of a finite-dimensional algebra*, J. Algebra**197**(1997), no. 2, 535–545. MR**1483779**, DOI https://doi.org/10.1006/jabr.1997.7140 - Azaria Paz,
*An application of the Cayley-Hamilton theorem to matrix polynomials in several variables*, Linear and Multilinear Algebra**15**(1984), no. 2, 161–170. MR**740668**, DOI https://doi.org/10.1080/03081088408817585 - Heydar Radjavi and Peter Rosenthal,
*Matrices for operators and generators of $B({\cal H})$*, J. London Math. Soc. (2)**2**(1970), 557–560. MR**265978**, DOI https://doi.org/10.1112/jlms/2.Part_3.557

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
15A30,
47L05

Retrieve articles in all journals with MSC (2010): 15A30, 47L05

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

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.