Title

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

Authors

Stephen P. Glasby, Central Washington University
Cheryl E. Praeger, University of Western Australia

Document Type

Article

Department or Administrative Unit

Mathematics

Publication Date

8-1-2009

Abstract

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 F_q, the proportion of matrices in M (n,F_q)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,F_q) 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,F_q).

Comments

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.

Due to copyright restrictions, this article is not available for free download from ScholarWorks @ CWU.

Recommended Citation

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

Journal of Algebra