The density of uncyclic matrices

TL;DR

This paper studies the proportion of 'good' matrices in the algebra of all matrices over finite fields, proving exponential decay of 'not good' matrices and providing bounds, conjectures, and an efficient testing algorithm.

Contribution

It establishes the exponential decay of 'not good' matrices' density, provides lower bounds for 'good' matrices, and introduces a Monte Carlo algorithm for testing matrix goodness.

Findings

Proportion of 'not good' matrices decays exponentially with dimension.

Lower bounds for the density of 'good' matrices depending on field size.

Efficient Monte Carlo algorithm for testing matrix 'goodness'.

Abstract

An element $X$ in the algebra ${\rm M}(n,\mathbb{F})$ of all $n\times n$ matrices over a field $\mathbb{F}$ is said to be $f$-cyclic if the underlying vector space considered as an $\mathbb{F}[X]$-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 MeatAxe algorithm. We prove that, for any finite field $\mathbb{F}_q$, the proportion of matrices in ${\rm M}(n,\mathbb{F}_q)$ 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 ${\rm M}(n,\mathbb{F}_q)$ for the MeatAxe depends on the degree, showing that it is at least $1-\frac2q(\frac{1}{q}+\frac{1}{q^2}+\frac{2}{q^3})^n$ for $q\geq4$. We conjecture that the density is at least $1-\frac1q(\frac{1}{q}+\frac{1}{2q^2})^n$ for all $q$ and $n$, and confirm this conjecture for dimensions $n\leq 37$. Finally we give a one-sided Monte Carlo algorithm called IsfCyclic to test whether a matrix is `good', at a cost of ${\rm O}({\rm Mat}(n)\log n)$ field operations, where ${\rm Mat}(n)$ is an upper bound for the number of field operations required to multiply two matrices in ${\rm M}(n,\mathbb{F}_q)$.