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).
Glasby, S. P., & Praeger, C. E. (2009). Towards an efficient Meat-axe algorithm using f-cyclic matrices: The density of uncyclic matrices in m(n,q). Journal of Algebra, 322(3), 766–790. https://doi.org/10.1016/j.jalgebra.2009.02.021
Journal of Algebra