Generalized derivatives of eigenvalues of a symmetric matrix

Abstract

Sensitivity theory is established for eigenvalues of a symmetric matrix depending on an arbitrary number of parameters; a sequence of symmetric eigenvalue problems is derived whose solution yields first-order generalized derivative information in the form of Clarke subgradients. This is accomplished using lexicographic directional differentiation, a recently developed tool in nonsmooth analysis. The findings here include sensitivities of multiple eigenvalues of symmetric matrices depending nonsmoothly on parameters and sensitivities of nonsmooth functions of symmetric matrix eigenvalues. The approach here is computationally relevant (with an algorithm given), and classical theory is recovered along the way (such as if an eigenvalue is simple). Thanks to the flexibility of this approach, the present theory is amenable to compositions of symmetric matrix eigenvalue problems with other problems, such as optimization or dynamic problems.