Volume 60, pp. 364-380, 2024.

Relative perturbation tanΘ-theorems for definite matrix pairs

Suzana Miodragović, Ninoslav Truhar, and Ivana Kuzmanović Ivičić

Abstract

In this paper, we consider perturbations of a Hermitian matrix pair $(H,M)$, where $H=GJ{G}^{\ast }$ is non-singular, $J=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\pm 1)$, and $M$ is a positive definite matrix. The corresponding perturbed pair defined as $(\tilde{H},\tilde{M})=(H+\delta H,M+\delta M)$ is such that $\tilde{H}=\tilde{G}J{\tilde{G}}^{\ast }$ is non-singular and $\tilde{M}$ is a positive definite matrix. An upper bound for the norm of the tangents of the angles between the eigenspaces of the perturbed and unperturbed pairs is derived. The rotation of the eigenspaces under a perturbation is measured in the scalar product induced by $M$. We show that a relative $\tan \mathrm{\Theta }$-bound for the standard eigenvalue problem is a special case of our new bound.

Full Text (PDF) [484 KB], BibTeX , DOI: 10.1553/etna_vol60s364

Key words

perturbation of matrix pairs, rotation of subspaces, tangent theta theorem, eigenvalues, eigenspaces

AMS subject classifications

65F99, 15A42, 65F15, 47A55

Links to the cited ETNA articles

[5]	Vol. 45 (2016), pp. 33-57 Luka Grubišić, Suzana Miodragović, and Ninoslav Truhar: Double angle theorems for definite matrix pairs