Generalized Wong sequences and their applications to Edmonds' problems

TL;DR

This paper introduces two polynomial-time algorithms for matrix rank and determinant problems over fields, extending Wong sequences to matrix spaces, and solving open questions in symbolic matrix analysis.

Contribution

The paper generalizes Wong sequences to matrix spaces and provides the first polynomial algorithms for specific variants of Edmonds' matrix problems.

Findings

Algorithms work over finite fields and rationals

Solve open problems for matrices spanned by rank-one and triangularizable matrices

Extend Wong sequences to pairs of matrix spaces

Abstract

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n by n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matrices in B, symbolic determinant identity testing (SDIT) is the question to decide whether there exists a nonsingular matrix in B. The constructive versions of these problems are asking to find a matrix of maximum rank, respectively a nonsingular matrix, if there exists one. Our first algorithm solves the constructive SMR when B is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when B is spanned by triangularizable matrices, but the triangularization is not given explicitly. Both algorithms work over finite fields of size at least n+1 and over the rational numbers, and the first algorithm actually solves (the non-constructive) SMR independently from the field size. Our main tool to obtain these results is to generalize Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces.