Relative Residual Bounds For The Eigenvalues of a Hermitian Semidefinite Matrix
Zlatko Drmač, Vjeran Hari · SIAM Journal on Matrix Analysis and Applications · 1997
Let H be a Hermitian matrix, X an orthonormal matrix, and $M=X^*HX$. Then the eigenvalues of M approximate some eigenvalues of H with an absolute error bounded by $ s{R}$, $R=HX-XM$. 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 of 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.