An Inequality for Selfadjoint Operators on a Hilbert Space

Abstract

An elementary inequality of use in testing convergence of eigenvector calculations is proven. If ${e_\lambda }$ is a unit eigenvector corresponding to an eigenvalue $\lambda$ of a selfadjoint operator $A$ on a Hilbert space $H$, then \[ {\left | {(g,{e_\lambda })} \right |^2} \leq \frac {{{{\left \| g \right \|}^2}{{\left \| {Ag} \right \|}^2} - {{(g,Ag)}^2}}}{{{{\left \| {(A - \lambda I)g} \right \|}^2}}}\] for all $g$ in $H$ for which $Ag \ne \lambda g$. Equality holds only when the component of $g$ orthogonal to ${e_\lambda }$ is also an eigenvector of $A$.