ISSN:
0945-3245
Keywords:
Mathematics Subject Classification (1991): 65F15, 58C40, 15A18, 90C31, 49M45
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s002110050109
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