On the lengths of irreducible pairs of complex matrices
Authors:
W. E. Longstaff and Peter Rosenthal
Journal:
Proc. Amer. Math. Soc. 139 (2011), 37693777
MSC (2010):
Primary 15A30; Secondary 47L05
Published electronically:
June 13, 2011
MathSciNet review:
2823023
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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
(35 #4242)
 2.
D.
Constantine and M.
Darnall, Lengths of finite dimensional representations of PBW
algebras, Linear Algebra Appl. 395 (2005),
175–181. MR 2112883
(2005i:17009), http://dx.doi.org/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 TausskyTodd: in memoriam. MR 1610851
(99b:20098), http://dx.doi.org/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
(2010i:15047), http://dx.doi.org/10.1017/S0004972709000112
 5.
W.
E. Longstaff, Burnside’s theorem: irreducible pairs of
transformations, Linear Algebra Appl. 382 (2004),
247–269. MR 2050111
(2005b:47159), http://dx.doi.org/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
(2007c:15024), http://dx.doi.org/10.1017/S0004972700035462
 7.
Christopher
J. Pappacena, An upper bound for the length of a finitedimensional
algebra, J. Algebra 197 (1997), no. 2,
535–545. MR 1483779
(98j:16013), http://dx.doi.org/10.1006/jabr.1997.7140
 8.
Azaria
Paz, An application of the CayleyHamilton theorem to matrix
polynomials in several variables, Linear and Multilinear Algebra
15 (1984), no. 2, 161–170. MR 740668
(85h:15019), http://dx.doi.org/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
(42 #887)
 1.
 L. Brickman and P. A. Fillmore, `The invariant subspace lattice of a linear transformation', Canad. J. Math. 19 (1967), 810822. MR 0213378 (35:4242)
 2.
 D. Constantine and M. Darnall, `Lengths of finite dimensional representations of PBW algebras', Linear Algebra Appl. 395 (2005), 175181. MR 2112883 (2005i:17009)
 3.
 A. Freedman, R. Gupta and R. Guralnick, `Shirshov's theorem and representations of semigroups', Pacific J. Math., Special Issue (1997), 159176. MR 1610851 (99b:20098)
 4.
 M. S. Lambrou and W. E. Longstaff, `On the lengths of pairs of complex matrices of size ', Bull. Aust. Math. Soc. 80 (2009), 177201. MR 2540352 (2010i:15047)
 5.
 W. E. Longstaff, `Burnside's Theorem: Irreducible pairs of transformations', Lin. Alg. & Applic. 382 (2004), 247269. MR 2050111 (2005b:47159)
 6.
 W. E. Longstaff, A. Niemeyer and Oreste Panaia, `On the lengths of pairs of complex matrices of size at most ', Bull. Aust. Math. Soc. 73 (2006), 461472. MR 2230653 (2007c:15024)
 7.
 Christopher J. Pappacena, `An upper bound for the length of a finitedimensional algebra', J. Algebra 197 (1997), 535545. MR 1483779 (98j:16013)
 8.
 A. Paz, `An application of the CayleyHamilton theorem to matrix polynomials in several variables', Lin. Mult. Algebra 15 (1984), 161170. MR 740668 (85h:15019)
 9.
 H. Radjavi and P. Rosenthal, `Matrices for operators and generators of ', J. London Math. Soc. (2) 2 (1970), 557560. MR 0265978 (42:887)
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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:
http://dx.doi.org/10.1090/S000299392011111493
PII:
S 00029939(2011)111493
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.
