An inequality for selfadjoint operators on a Hilbert space
Herbert Jacob Bernstein · Proceedings of the American Mathematical Society · 1987
An elementary inequality of use in testing convergence of eigenvector calculations is proven. If e λ {e_\lambda } is a unit eigenvector corresponding to an eigenvalue λ \lambda of a selfadjoint operator A A on a Hilbert space H H , then \[ | ( g , e λ ) | 2 ≤ ‖ g ‖ 2 ‖ A g ‖ 2 − ( g , A g ) 2 ‖ ( A − λ I ) g ‖ 2 {\left | {(g,{e_\lambda })} \right |^2} \leq \frac {{{{\left \| g \right \|}^2}{{\left \| {Ag} \right \|}^2} - {{(g,Ag)}^2}}}{{{{\left \| {(A - \lambda I)g} \right \|}^2}}} \] for all g g in H H for which A g ≠ λ g Ag e \lambda g . Equality holds only when the component of g g orthogonal to e λ {e_\lambda } is also an eigenvector of A A .