Perturbations of the eigenprojections of a factorized Hermitian matrix
Perturbation bounds are given for the eigenprojections of a Hermitian matrix \(H= GJG^*\), where \(G\) is an \(n\times r\) matrix of full column rank and \(J\) is a nonsingular Hermitian matrix, under the perturbation of the factor \(G\). The perturbed matrix \(H'\) is defined by \(H'= (G+ \delta G) J(G+ \delta G)^*= G' J(G')^*\), where \(\| \delta Gx\|_ 2\leq \eta\| Gx\|_ 2\). With the spectral projection belonging to a (possibly multiple) eigenvalue \(\lambda\) of \(H\) given by \(P={1\over 2\pi i} \int_ \Gamma (\mu I- H)^{-1} d\mu\), where \(\Gamma\) is a curve around \(\lambda\) which separates \(\lambda\) from the rest of the spectrum of \(H\), the main result is the following Theorem: Let \(\lambda\) be a positive (possibly multiple) eigenvalue of a Hermitian matrix \(H= GJG^*\), \(P\) be the corresponding eigenprojection and \(P'\) be the corresponding spectral projection of the perturbed matrix \(H'\). Then \[ \| P'- P\|_ 2\leq {4\over \text{rg} (\lambda)}\cdot {1\over 1- 3\overline\eta/ \text{rg} (\lambda)}, \] where \(\overline\eta= \eta(2+ \eta)\), \(\text{rg} (\lambda)= \min\{1, (\lambda_ R- \lambda)/(\lambda_ R+ \lambda), (\lambda- \lambda_ L)/(\lambda+ \lambda_ L)\}\), \(\lambda_ L\) and \(\lambda_ R\) denote the left and the right neighbor of \(\lambda\) in the spectrum \(\sigma(H)\), provided the right-hand side in the inequality is positive. The perturbation bounds for the eigenvector corresponding to simple eigenvalues are also given.
scientific article; zbMATH DE number 753875