Truhar, Ninoslav; Matematični kolokvij 2010

The rotation of eigenspaces of perturbed matrix pairs

Ninoslav Truhar

Univerza v Osijeku, Hrvaška

4. november 2010

We revisit the relative perturbation theory for invariant subspaces of positive definite matrix pairs. As a prototype example for our results we consider parameter dependent families of eigenvalue problems. We show that new estimates are a natural way to obtain sharp as functions of the parameter indexing the family of matrix pairs estimates for the rotation of spectral subspaces. We also present the upper bound for the norm of J-unitary matrix F (F * JF = J), which plays important role in the relative perturbation theory for quasi-definite Hermitian matrices H, where

$H_{qd} \equiv P^T H P = \begin{bmatrix} H_{11} & H_{12}\\ H^*_{12} & -H_{22} \end{bmatrix}, \qquad J = \begin{bmatrix} I_k & 0 \\ 0 & -I_{n-k} \end{bmatrix},$

for some permutation matrix P and $H_{11} \in \mathbb{C}^{k \times k}$ and $H_{22} + H^*_{12} H^{-1}_{11} H_{12} \in \mathbb{C}^{n-k\times n - k}$ positive definite.