A Bound for the Matrix Square Root with Application to Eigenvector Perturbation

Abstract

Let $H$ be positive definite, let $\Delta H$ be Hermitian, and suppose that $\|H^{-1/2} (\Delta H) H^{-1/2} \| =1$. It is shown that the least constant $c_n$ such that \[ || (H + \eta \Delta H)^{1/2}H^{-1/2} - I || \leq c_n \eta + 0(\eta^2) \] for all $n \times n$ $H$ and $\Delta H$ grows like $\log n$. This fact has consequences in eigenvector perturbation theory.