Total derivatives of eigenvalues and eigenprojections of symmetric matrices

Abstract

Conditions for existence and formulas for the first- and second order total derivatives of the eigenvalues, and the first order total derivatives of the eigenprojections of smooth matrix-valued functions $H\colon\Omega\to S(m)$ are given. The eigenvalues and eigenprojections are considered as functions in the same domain $\Omega\subseteq\mathbb{R}^n$.