The rotation of eigenspaces of perturbed matrix pairs

We present new sinΘ theorems for perturbations of positive definite matrix pairs. The rotation of eigenspaces is measured in the matrix dependent scalar product. We 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 sinΘ theorems for eigenspaces of positive definite Hermitian matrices are included as special cases in our approach. Our experiments indicate that relative sinΘ 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.