Sensitivity analysis of all eigenvalues of a symmetric matrix

Summary.

Given $A(x)= [a_{ij} (x)]$ a $n$-by-$n$ symmetric matrix depending (smoothly) on a parameter $x$, we study the first order sensitivity of all the eigenvalues$\lambda_m (x)$ of $A(x)$, $1\le m\le n$. Under some smoothness assumption like the $a_{ij}$ be $C^1$, we prove that the directional derivatives$d\mapsto \lambda^\prime_m (x,d) = \lim_{t \to o^+} [\lambda_m (x + td) - \lambda_m (x)] / t$ do exist and give an explicit expression of them in terms of the data of the parametrized matrix. The key idea to circumvent the difficulties inherent to the study of each$\lambda_m$ taken separately, is to consider the functions$f_m (x)$ , $1\le m\le n$, defined as the sums of the$m$ largest eigenvalues of $A(x)$. Based on Ky Fan's variational formulation of $f_m$ and some chain rule from nonsmooth analysis, we derive an explicit formula for the generalized gradient of $f_m$ and a computationally useful formula for the directional derivative of $f_m$. Using these formulas and the relation $\lambda_m= f_m - f_{m-1}$, we then derive the directional derivative of $\lambda_m$. Some properties of this directional derivative as well as an illustrative example are presented.