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

Published electronically:
June 13, 2011

MathSciNet review:
2823023

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The length of a pair of matrices is the smallest integer such that words in the matrices with at most factors span the unital algebra generated by the pair. Upper bounds for lengths have been much studied. If is a rank one (complex) matrix, the length of the irreducible pair is and the subwords of form a basis for . New examples are given of irreducible pairs of matrices of length . There exists an irreducible pair of matrices of length . We begin the study of determining lower bounds for lengths.

**1.**L. Brickman and P. A. Fillmore,*The invariant subspace lattice of a linear transformation*, Canad. J. Math.**19**(1967), 810–822. MR**0213378****2.**D. Constantine and M. Darnall,*Lengths of finite dimensional representations of PBW algebras*, Linear Algebra Appl.**395**(2005), 175–181. MR**2112883**, 10.1016/j.laa.2004.08.006**3.**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**, 10.2140/pjm.1997.181.159**4.**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**, 10.1017/S0004972709000112**5.**W. E. Longstaff,*Burnside’s theorem: irreducible pairs of transformations*, Linear Algebra Appl.**382**(2004), 247–269. MR**2050111**, 10.1016/j.laa.2003.12.043**6.**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**, 10.1017/S0004972700035462**7.**Christopher J. Pappacena,*An upper bound for the length of a finite-dimensional algebra*, J. Algebra**197**(1997), no. 2, 535–545. MR**1483779**, 10.1006/jabr.1997.7140**8.**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**, 10.1080/03081088408817585**9.**Heydar Radjavi and Peter Rosenthal,*Matrices for operators and generators of 𝐵(\cal𝐻)*, J. London Math. Soc. (2)**2**(1970), 557–560. MR**0265978**

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

DOI:
http://dx.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.