On the lengths of irreducible pairs of complex matrices

Longstaff, W.; Rosenthal, Peter

doi:10.1090/S0002-9939-2011-11149-3

On the lengths of irreducible pairs of complex matrices
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by W. E. Longstaff and Peter Rosenthal PDF

Proc. Amer. Math. Soc. 139 (2011), 3769-3777 Request permission

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.

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