Canonical forms for matrix tuples in polynomial time

TL;DR

This paper introduces polynomial-time algorithms for computing canonical forms of matrix tuples over finite fields under specific group actions, advancing the understanding of their structural properties and applications.

Contribution

It provides the first polynomial-time algorithms for canonical forms of matrix tuples under left-right and conjugation actions, based on new structural insights.

Findings

Algorithms run in polynomial time

Structural insights generalize Schur's lemma

Applicable to problems in polynomial identity testing and group isomorphism

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.