Symbolic determinant identity testing and non-commutative ranks of matrix Lie algebras

TL;DR

This paper investigates matrix Lie algebras to understand singular matrix spaces with full non-commutative rank, providing polynomial-time algorithms for symbolic determinant identity testing and characterizing when such spaces have singularity certificates.

Contribution

It establishes that complex matrix Lie algebras produce singular matrix spaces with full non-commutative rank and offers a deterministic polynomial-time method for SDIT of these spaces.

Findings

Matrix Lie algebras over complex numbers yield singular spaces with full non-commutative rank.

SDIT for these spaces can be decided in deterministic polynomial time.

Characterization of Lie algebras that produce spaces with singularity certificates.

Abstract

One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose non-commutative rank is full. At present, examples of such matrix spaces are mostly sporadic, so it is desirable to discover them in a more systematic way. In this paper, we make a step towards this direction, by studying the family of matrix spaces that are closed under the commutator operation, that is matrix Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the complex number field give rise to singular matrix spaces with full non-commutative ranks. On the other hand, we show that SDIT of such spaces can be decided in deterministic polynomial time. Moreover, we give a characterization for the matrix Lie algebras to yield a matrix space possessing singularity certificates as studied by Lov'asz (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).