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An Exact Algorithm for Computing the Structure of Jordan Blocks

Shinichi Tajima, Katsuyoshi Ohara, Akira Terui

TL;DR

This work addresses the problem of determining the Jordan block structure of a matrix over the rationals without computing complete Jordan chains. It introduces the Jordan-Krylov framework, computing a seed from $\ker f(A)^{\bar{\ell}}$ and deriving the entire structure from an extended Krylov generating set, while staying in $\mathbb{Q}$ and avoiding linear system solves over extensions. The authors provide an algorithm that builds a Krylov generating set, extends it, and performs Jordan-Krylov elimination to obtain the counts $c_i$ of blocks of each length, with a complexity that is competitive vs naive approaches and theoretically favorable when many irreducible factors are present. Empirical results demonstrate significant speedups from preprocessing and matrix-form implementations, especially in cases with multiple irreducible factors, confirming the practical value of focusing on structure rather than full Jordan chains.

Abstract

An efficient method is proposed for computing the structure of Jordan blocks of a matrix of integers or rational numbers by exact computation. We have given a method for computing Jordan chains of a matrix with exact computation. However, for deriving just the structure of Jordan chains, the algorithm can be reduced to increase its efficiency. We propose a modification of the algorithm for that purpose. Results of numerical experiments are given.

An Exact Algorithm for Computing the Structure of Jordan Blocks

TL;DR

This work addresses the problem of determining the Jordan block structure of a matrix over the rationals without computing complete Jordan chains. It introduces the Jordan-Krylov framework, computing a seed from and deriving the entire structure from an extended Krylov generating set, while staying in and avoiding linear system solves over extensions. The authors provide an algorithm that builds a Krylov generating set, extends it, and performs Jordan-Krylov elimination to obtain the counts of blocks of each length, with a complexity that is competitive vs naive approaches and theoretically favorable when many irreducible factors are present. Empirical results demonstrate significant speedups from preprocessing and matrix-form implementations, especially in cases with multiple irreducible factors, confirming the practical value of focusing on structure rather than full Jordan chains.

Abstract

An efficient method is proposed for computing the structure of Jordan blocks of a matrix of integers or rational numbers by exact computation. We have given a method for computing Jordan chains of a matrix with exact computation. However, for deriving just the structure of Jordan chains, the algorithm can be reduced to increase its efficiency. We propose a modification of the algorithm for that purpose. Results of numerical experiments are given.

Paper Structure

This paper contains 13 sections, 6 theorems, 15 equations, 5 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. The generalized eigenspace of a matrix and a Jordan-Krylov basis of $\ker f(A)^{{\bar{\ell}}}$
  3. An algorithm for computing the structure of Jordan blocks
  4. Calculating Krylov generating set
  5. Computing the structure of Jordan blocks
  6. Complexity analysis
  7. Experiments
  8. A matrix with a Jordan blocks of size $1$ and $4$
  9. A matrix with a Jordan blocks of size $2$ and $10$
  10. A matrix with a Jordan blocks of size $6$
  11. A matrix with more than one irreducible factor in the characteristic polynomial
  12. A matrix with many irreducible factors in the characteristic polynomial
  13. Concluding remarks

Key Result

Theorem 2

$\ker f(A)^{{\bar{\ell}}}$ has a Jordan-Krylov basis.

Theorems & Definitions (9)

  • Definition 1: Krylov generating set, Jordan-Krylov basis (taj-oha-ter2022)
  • Theorem 2: taj-oha-ter2022
  • Theorem 3
  • Theorem 4: taj-oha-ter2022
  • Definition 5: The minimal annihilating polynomial
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Remark 1