2017/11/01 by Yuji Nakatsukasa, Vanni Noferini, Nakatsukasa, Yuji +1 · 1 citation
Computer Science · Engineering · Mathematics · #15A22 (Secondary) #65F15 (Primary) 15A18 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #VLSI and FPGA Design Techniques
paper · pdf · doi:10.48550/arxiv.1711.00495
openalex publication_date 2017/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems.