Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Canonical forms for matrix tuples in polynomial time

Youming Qiao, Xiaorui Sun

TL;DR

This work develops polynomial-time Las-Vega algorithms to compute canonical forms for matrix tuples over finite fields under left-right and conjugation actions. It introduces a structural framework based on indecomposable-block-corresponding row-submatrix (IBC) tuples, hierarchical row-tuple decompositions, and quotient matrix tuples to reveal block structure and enable canonical representations. The core contribution is a general canonical-form algorithm built from detailed subroutines (IBC-tuple selection, direct-sum decomposition, essential extensions, compression matrices) that operate canonically across equivalent inputs, with a separate treatment for full-rank square tuples under conjugation. The framework links matrix-tuple canonical forms to tensor canonical forms and group isomorphism problems, offering a tractable approach to otherwise wild classification problems and providing a foundation for practical orbit-closure and tensor-isomorphism analyses in finite-field settings.

Abstract

Left-right and conjugation actions on matrix tuples have received considerable attention in theoretical computer science due to their connections with polynomial identity testing, group isomorphism, and tensor isomorphism. In this paper, we present polynomial-time algorithms for computing canonical forms of matrix tuples over a finite field under these actions. Our algorithm builds upon new structural insights for matrix tuples, which can be viewed as a generalization of Schur's lemma for irreducible representations to general representations.

Canonical forms for matrix tuples in polynomial time

TL;DR

This work develops polynomial-time Las-Vega algorithms to compute canonical forms for matrix tuples over finite fields under left-right and conjugation actions. It introduces a structural framework based on indecomposable-block-corresponding row-submatrix (IBC) tuples, hierarchical row-tuple decompositions, and quotient matrix tuples to reveal block structure and enable canonical representations. The core contribution is a general canonical-form algorithm built from detailed subroutines (IBC-tuple selection, direct-sum decomposition, essential extensions, compression matrices) that operate canonically across equivalent inputs, with a separate treatment for full-rank square tuples under conjugation. The framework links matrix-tuple canonical forms to tensor canonical forms and group isomorphism problems, offering a tractable approach to otherwise wild classification problems and providing a foundation for practical orbit-closure and tensor-isomorphism analyses in finite-field settings.

Abstract

Left-right and conjugation actions on matrix tuples have received considerable attention in theoretical computer science due to their connections with polynomial identity testing, group isomorphism, and tensor isomorphism. In this paper, we present polynomial-time algorithms for computing canonical forms of matrix tuples over a finite field under these actions. Our algorithm builds upon new structural insights for matrix tuples, which can be viewed as a generalization of Schur's lemma for irreducible representations to general representations.
Paper Structure (51 sections, 23 theorems, 194 equations)

This paper contains 51 sections, 23 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Motivations from theoretical computer science
  4. Orbit closure intersection problems.
  5. Matrix tuples from group isomorphism.
  6. Towards understanding tensor isomorphism and canonical form.
  7. Motivations from mathematics
  8. Matrix tuples as a wild classification problem.
  9. Isomorphism and canonical form algorithms.
  10. Main results
  11. Overview
  12. Matrix tuple without non-trivial characteristic row tuple subspace
  13. Matrix tuple with direct sum row tuple decomposition
  14. Matrix tuple with hierarchical row tuple decomposition
  15. Hierarchical row tuple decomposition and quotient matrix tuple
  16. ...and 36 more sections

Key Result

Theorem 1.1

There is a randomized Las-Vegas algorithm to compute a canonical form of a matrix tuple in $\mathrm{M}(n\times m, \mathbb{F}_q)^\ell$ under the equivalence relation in $\mathrm{poly}(n, m, \ell, \log q)$ time.

Theorems & Definitions (106)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 3.2: Ber67CZ81
  • Proposition 3.3: IR99
  • Theorem 3.4: FR85Iva00Ron90
  • Lemma 4.1: Kir16
  • proof
  • Definition 4.3: Indecomposable-Block-Corresponding Row-Submatrix Tuple
  • Definition 4.4
  • ...and 96 more