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ON THE LENGTHS OF PAIRS OF COMPLEX MATRICES OF SIZE SIX

  • M. S. LAMBROU (a1) and W. E. LONGSTAFF (a2)

Abstract

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.

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Copyright

Corresponding author

For correspondence; e-mail: longstaf@maths.uwa.edu.au

References

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[1] Constantine, D. and Darnall, M., ‘Lengths of finite dimensional representations of PBW algebras’, Linear Algebra Appl. 395 (2005), 175181.
[2] Freedman, A., Gupta, R. and Guralnick, R., ‘Shirshov’s theorem and representations of semigroups’, Pacific J. Math., Special Issue (1997), 159176.
[3] Longstaff, W. E., ‘Burnside’s theorem: irreducible pairs of transformations’, Linear Algebra Appl. 382 (2004), 247269.
[4] Longstaff, W. E., ‘On words in a,b with more than 36 subwords’, unpublished manuscript (2005).
[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), 461472.
[6] Paz, A., ‘An application of the Cayley–Hamilton theorem to matrix polynomials in several variables’, Linear Multilinear Algebra 15 (1984), 161170.
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Keywords

MSC classification

ON THE LENGTHS OF PAIRS OF COMPLEX MATRICES OF SIZE SIX

  • M. S. LAMBROU (a1) and W. E. LONGSTAFF (a2)

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