Numerische Mathematik

, Volume 70, Issue 1, pp 45–72

# Sensitivity analysis of all eigenvalues of a symmetric matrix

• J.-B. Hiriart-Urruty
• D. Ye

## 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.

Mathematics Subject Classification (1991): 65F15, 58C40, 15A18, 90C31, 49M45