Inequalities for generalized eigenvalues of quaternion matrices (Q2043723)

The paper deals with a particular generalized eigenvalue problem. It concerns a pair \(A,B\) of self-conjugate square matrices, where \(B\) is a positive definite matrix. Equivalently, this is an eigenvalue problem for the matrix \(B^{-1}A\) (see [\textit{G. W. Stewart}, Linear Algebra Appl. 390, 1--5 (2004; Zbl 1067.15016)]), strictly connected with the eigenvalue problem for Hermitian matrix pencils (see [\textit{C.-K. Li} and \textit{R. Mathias}, Linear Algebra Appl. 271, 309--321 (1998; Zbl 0890.15007)]). In the case of quaternion matrices, there are additional assumptions and questions because of the non-commutative multiplication (left and right cases). In this paper, some results from the Hermitian environment are reproved in a more general setting with quaternion matrices. The main results concern extremal properties of generalized eigenvalues based on modified Rayleigh-quotients and eigenvalues bounds after additive perturbations of \(A\) or \(B\) (see [\textit{Y. Hong} et al., J. Inequal. Appl. 2018, Paper No. 155, 6 p. (2018; Zbl 1498.15022)]).