Nilpotent-independent sets and estimation in matrix algebras

TL;DR

This paper presents a new method for estimating the proportion of matrices with certain properties in matrix algebras over finite fields, aiding the analysis of randomized algorithms in computational linear algebra.

Contribution

It introduces a novel approach to estimate proportions of matrix families based on their invertible parts, extending existing techniques to include singular matrices.

Findings

Effective estimation of matrix family proportions depending on invertible parts.

Application to primary cyclic matrices in the Holt-Rees MEAT-AXE algorithm.

Enhanced analysis of randomized algorithms for matrix computations.

Abstract

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for these algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families $N$ of elements in the algebra of all $d\times d$ matrices over a field of order $q$, where membership of a matrix in $N$ depends only on its `invertible part'. The method is based on estimating proportions of certain subsets of $\text{GL}(d,q)$ depending on $N$, so that existing estimation techniques for nonsingular matrices can be leveraged to deal with families containing singular matrices. As an application we investigate primary cyclic matrices, which are used in the Holt-Rees MEAT-AXE algorithm for testing irreducibility of matrix algebras.