RELATIVE RESIDUAL BOUNDS FOR THE EIGENVALUES OF A HERMITIAN SEMIDEFINITE MATRIX

Let H be a Hermitian matrix, X an orthonormal matrix, and M = XHX. Th en the eigenvalues of M approximate some eigenvalues of H with an abso lute error bounded by parallel to R parallel to(2), R = HX - XM. This work contains estimates of \lambda-mu\/\mu\ and \lambda-mu\/\lambda\, where mu, lambda is a matching pair of the eigenvalues of M and H when H is semidefinite. The general bound is expressed in terms of sines o f the canonical angles between certain subspaces associated with H and X. A. more refined quadratic bound which uses the relative distances between eigenvalues is also proved.