Computing Minimal Polynomials of Matrices

TL;DR

This paper introduces a Monte-Carlo algorithm for efficiently computing the minimal polynomial of matrices over finite fields, with practical implementation and verification methods.

Contribution

It presents a new Monte-Carlo algorithm with complexity analysis, a deterministic verification procedure, and experimental comparisons with existing algorithms.

Findings

Algorithm requires O(n^3) field operations

Verification procedure has worst-case complexity of O(n^4)

Experimental results show competitive performance in practice

Abstract

We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an $n\times n$ matrix over a finite field that requires $O(n^3)$ field operations and O(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard algorithms for polynomial and matrix operations. We compare features of the algorithm with several other algorithms in the literature. In addition we present a deterministic verification procedure which is similarly efficient in most cases but has a worst-case complexity of $O(n^4)$. Finally, we report the results of practical experiments with an implementation of our algorithms in comparison with the current algorithms in the {\sf GAP} library.