On residual bounds for eigenvalues (Q1209771)

Given an \(n\times n\) Hermitian matrix \(A=\bigl[ {M \atop R} {{R^*} \atop N}\bigr]\), where \(M\) is a \(k\times k\) matrix, and let \(\lambda_ 1\geq\lambda_ 2 \geq\dots\geq\lambda_ n\) and \(\mu_ 1\geq \mu_ 2\geq\dots\geq \mu_ k\) be the eigenvalues of \(A\) and \(M\), respectively. If \(\| T\|\) denotes the spectral norm (the operator bound norm) of the matrix \(T\) one proves that \(\|\text{diag}(\mu_ 1-\lambda_{i_ 1}, \dots,\mu_ k-\lambda_{i_ k})\| \leq \| R\|\), but this does not hold for all unitarily invariant norms.