Towards an efficientMeat-axealgorithm usingf-cyclicmatrices: The density of uncyclic matrices in M(n,q)

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An element X in the algebra M(n,F) of all n × n matrices over a field F is said to be f-cyclic if the underlying vector space considered as an F[X]-module has at least one cyclic primary component. These are the matrices considered to be “good” in the Holt–Rees version of Norton’s irreducibility test in the Meat-axe algorithm. We prove that, for any finite field Fq, the proportion of matrices in M (n,Fq)that are “not good” decays exponentially to zero as the dimension n approaches infinity. Turning this around, we prove that the density of “good” matrices in M(n,Fq) for the Meat-axe depends on the degree, showing that it is at least 1− (2/q) ((1/q)+(1/q2)+(2/q3))n for q ≥ 4. We conjecture that the density is at least 1−(1/q)((1/q)+(1/2q2))n for all q and n, and confirm this conjecture for dimensions n ≤ 37. Finally we give a one-sided Monte Carlo algorithm called Is f Cyclic to test whether a matrix is “good,” at a cost of O(Mat(n) log n) field operations, where Mat(n) is an upper bound for the number of field operations required to multiply two matrices in M(n,Fq).


This article was originally published in Journal of Algebra with an Elsevier user license. The full-text article from the publisher can be found here.

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Journal of Algebra