Exact Algorithms for Computing Generalized Eigenspaces of Matrices via Jordan-Krylov Basis

TL;DR

The paper introduces an exact algorithm leveraging Jordan-Krylov basis and minimal annihilating polynomials to compute generalized eigenspaces of matrices with integer or rational entries, producing Jordan chains with polynomial components.

Contribution

It presents a novel Jordan-Krylov elimination method for efficiently computing generalized eigenspaces and Jordan chains with polynomial eigenvector components.

Findings

Algorithm accurately computes generalized eigenspaces.

Outputs include Jordan chains with polynomial components.

Method is effective for integer and rational matrices.

Abstract

An effective exact method is proposed for computing generalized eigenspaces of a matrix of integers or rational numbers. Keys of our approach are the use of minimal annihilating polynomials and the concept of the Jourdan-Krylov basis. A new method, called Jordan-Krylov elimination, is introduced to design an algorithm for computing Jordan-Krylov basis. The resulting algorithm outputs generalized eigenspaces as a form of Jordan chains. Notably, in the output, components of generalized eigenvectors are expressed as polynomials in the associated eigenvalue as a variable.