eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-01-25
87:1
87:21
10.4230/LIPIcs.ITCS.2022.87
article
Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras
Ivanyos, Gábor
1
https://orcid.org/0000-0003-3826-1735
Mittal, Tushant
2
https://orcid.org/0000-0002-4017-2662
Qiao, Youming
3
https://orcid.org/0000-0003-4334-1449
Institute for Computer Science and Control, Eötvös Loránd Research Network (ELKH), Budapest, Hungary
Department of Computer Science, University of Chicago, IL, USA
Centre for Quantum Software and Information, University of Technology Sydney, Australia
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 Lovász (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol215-itcs2022/LIPIcs.ITCS.2022.87/LIPIcs.ITCS.2022.87.pdf
derandomization
polynomial identity testing
symbolic determinant
non-commutative rank
Lie algebras