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

  • W. E. Longstaff (a1), A. C. Niemeyer (a1) and Oreste Panaia (a1)

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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.

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References

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[1]Freedman, A., Gupta, R. and Guralnick, R., ‘Shirshov's theorem and representations of semigroups’, Pacific J. Math., Special Issue (1997), 159176.
[2]Longstaff, W.E., ‘Burnside's Theorem: irreducible pairs of transformations’, Linear Algebra Appl. 382 (2004), 247269.
[3]Pappacena, C.J., ‘An upper bound for the length of a finite-dimensional algebra’, J. Algebra 197 (1997), 535545.
[4]Paz, A., ‘An application of the Cayley-Hamilton theorem to matrix polynomials in several variables’, Linear and Multilinear Algebra 15 (1984), 161170.
[5]Procesi, C., Rings with polynomial identities (Marcel Dekker, New York, 1973).
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On the lengths of pairs of complex matrices of size at most five

  • W. E. Longstaff (a1), A. C. Niemeyer (a1) and Oreste Panaia (a1)

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