Quadratic Residual Bounds for the Hermitian Eigenvalue Problem

Let \[ A = \left[ \begin{array}{cc} M & R \\ R^{\ast} & N \end{array} \right] {\rm and } \tilde{A} = \left[ \begin{array}{cc} M & 0 \\ 0 & N \end{array} \right] \] be Hermitian matrices. Stronger and more general $O(\|R\|^2)$ bounds relating the eigenvalues of A and A are proved using a Schur complement technique. These results extend to singular values, to eigenvalues of non-Hermitian matrices, and to generalized eigenvalues.