Lehmann Bounds and Eigenvalue Error Estimation
SIAM Journal on Numerical Analysis
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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 $\ns{R}$, $R=HX-XM$. 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.