Symbolic determinant identity testing and noncommutative ranks of matrix Lie algebras
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 noncommutative rank problem (GargGurvitsOliveiraWigderson, Found. Comput. Math. 2020; IvanyosQiaoSubrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose noncommutative 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 noncommutative 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).
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.06403
 Bibcode:
 2021arXiv210906403I
 Keywords:

 Computer Science  Computational Complexity;
 Mathematics  Representation Theory
 EPrint:
 23 pages