Fast Algorithm for Calculating the Minimal Annihilating Polynomials of Matrices via Pseudo Annihilating Polynomials

TL;DR

This paper introduces a fast algorithm for computing minimal annihilating polynomials of matrices using pseudo annihilating polynomials and binary splitting, improving efficiency in exact linear algebra computations.

Contribution

It presents a novel method that efficiently calculates minimal annihilating polynomials for all unit vectors, leveraging pseudo annihilating polynomials and binary splitting techniques.

Findings

Algorithm has improved arithmetic time complexity

Efficient for matrices over fields of characteristic zero

Applicable to all unit vectors in the matrix space

Abstract

Minimal annihilating polynomials are very useful in a wide variety of algorithms in exact linear algebra. A new efficient method is proposed for calculating the minimal annihilating polynomials for all the unit vectors, for a square matrix over a field of characteristic zero. Key ideas of the proposed method are the concept of pseudo annihilating polynomial and the use of binary splitting technique. Efficiency of the resulting algorithms is shown by arithmetic time complexity analysis.