Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices
Applied Numerical Mathematics
Lehmann Bounds and Eigenvalue Error Estimation
SIAM Journal on Numerical Analysis
dqds with Aggressive Early Deflation
SIAM Journal on Matrix Analysis and Applications
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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 à are proved using a Schur complement technique. These results extend to singular values, to eigenvalues of non-Hermitian matrices, and to generalized eigenvalues.