# Sensitivity analysis of all eigenvalues of a symmetric matrix

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## 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.

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