The rotation of eigenspaces of perturbed matrix pairs
The authors consider the generalized eigenvector problem \(Hx=\lambda Mx\) and the corresponding perturbed one \((H+ \delta H)\tilde x =\tilde \lambda(M+\delta M)\tilde x\), where \(H, M, H+ \delta H, M+\delta M\in {\mathbb C}_{n\times n}\) are positive definite matrices. They present new \(\sin \Theta\) theorems for perturbations of positive definite matrix pairs. The rotation of eigenspaces is measured in the matrix dependent scalar product. They assess the sharpness of the new estimates in terms of effectivity quotients (the quotient of the measure of the perturbation and the estimator). Known relative \(\sin \Theta\) theorems for eigenspaces of positive definite Hermitian matrices are then special cases. Numerical experiments indicate that relative \(\sin \Theta\) theorems are asymptotically sharp when the rotation is measured in the appropriate matrix dependent scalar product but not always in the ordinary Euclidean scalar product.
scientific article; zbMATH DE number 6033312
scientific article