학술논문
The complexity of the inertia.
Document Type
Proceedings Paper
Author
Hoang, Thanh Minh (D-ULM-TI) AMS Author Profile; Thierauf, Thomas (D-AALEN-ELI) AMS Author Profile
Source
Subject
15 Linear and multilinear algebra; matrix theory -- 15A Basic linear algebra
15A18Eigenvalues, singular values, and eigenvectors
68Computer science -- 68Q Theory of computing
68Q15Complexity classes
15A18
68
68Q15
Language
English
Abstract
Summary: ``The inertia of a square matrix $A$ is defined as the triple $(i_+(A),i_-(A),i_0(A))$, where $i_+(A),i_-(A)$, and $i_0(A)$ are the number of eigenvalues of $A$, counting multiplicities, with positive, negative, and zero real part, respectively. A hard problem in linear algebra is to compute the inertia. There is no known method to do this exactly, in general. In this paper we show that the inertia is hard for PL (probabilistic logspace) and in some cases the inertia can be computed in PL. We extend our result to some problems related to the inertia. Namely, we show that matrix stability is complete for PL and the inertia of symmetric matrices can be computed in PL.''