A sin 2 Theorem for Graded Inde nite Hermitian Matrices 1

This paper gives double angle theorems that bound the change in an invariant subspace of an indeenite Hermitian matrix in the graded form H = D AD subject to a perturbation H ! e H = D (A + A)D. These theorems extend recent results on a deenite Hermitian matrix in the graded form (Linear Algebra Appl., 311 (2000), 45{60) but the bounds here are more complicated in that they depend on not only relative gaps and norms of A as in the deenite case but also norms of so-called the hyperbolic eigenvector matrices of certain associated matrix pairs. For two special but interest cases, bounds on these hyperbolic eigenvector matrices are obtained to show that their norms are of moderate magnitude. Abstract This paper gives double angle theorems that bound the change in an invariant subspace of an indeenite Hermitian matrix in the graded form H = D AD subject to a perturbation H ! e H = D (A + A)D. These theorems extend recent results on a deenite Hermitian matrix in the graded form (Linear Algebra Appl., 311 (2000), 45{60) but the bounds here are more complicated in that they depend on not only relative gaps and norms of A as in the deenite case but also norms of so-called the hyperbolic eigenvector matrices of certain associated matrix pairs. For two special but interest cases, bounds on these hyperbolic eigenvector matrices are obtained to show that their norms are of moderate magnitude.