Relative perturbation theory for definite matrix pairs and hyperbolic eigenvalue problem

In this paper, new relative perturbation bounds for the eigenvalues as well as for the eigensubspaces are developed for definite Hermitian matrix pairs and the quadratic hyperbolic eigenvalue problem. First, we derive relative perturbation bounds for the eigenvalues and the sin⁡Θsin⁡Θ type theorems for the eigensubspaces of the definite matrix pairs (A,B)(A,B), where both A,B∈Cm×mA,B∈Cm×m are Hermitian nonsingular matrices with particular emphasis, where B   is a diagonal of ±1. Further, we consider the following quadratic hyperbolic eigenvalue problem (μ2M+μC+K)v=0(μ2M+μC+K)v=0, where M,C,K∈Cn×nM,C,K∈Cn×n are given Hermitian matrices. Using proper linearization and new relative perturbation bounds for definite matrix pairs (A,B)(A,B), we develop corresponding relative perturbation bounds for the eigenvalues and the sin⁡Θsin⁡Θ type theorems for the eigensubspaces for the considered quadratic hyperbolic eigenvalue problem. The new bounds are uniform and depend only on matrices M, C, K, perturbations δM, δC and δK and standard relative gaps. The quality of new bounds is illustrated through numerical examples.