Variations of Ritz and Lehmann Bounds

Eigenvalue estimates that are optimal in some sense have self-evident appeal and leave estimators with a sense of virtue and economy. So, it is natural that ongoing searches for effective strategies for difficult tasks such as estimating matrix eigenvalues that are situated well into the interior of the spectrum revisit from time to time methods that are known to yield optimal bounds. This article reviews a variety of results related to obtaining optimal bounds to matrix eigenvalues --- some results are well-known; others are less known; and a few are new. We focus especially on Ritz and harmonic Ritz values, and right- and left-definite variants of Lehmann's method.