Two simple residual bounds for the eigenvalues of a Hermintian matrix

Let A be Hermitian and let the orthonormal columns of X span an approximate invariant subspace of X. Then the residual $R = AX - XM\, ( M = X^H AX )$ will be small. The theorems of this paper bound the distance of the spectrum of M from the spectrum of A in terms of appropriate norms of R.