On analyticity of functions involving eigenvalues

Abstract

Let A( z) be an n × n complex matrix whose elements depend analytically on z ∈ C m . It is well known that any individual eigenvalue of A( z) may be nondifferentiable when it coalesces with others. In this paper, we investigate the analycity property of functions on the eigenvalues λ( z) = ( λ 1( z),…, λ n ( z)) of A( z). We first introduce the notion of functions that are symmetric with respect to partitions. It is then shown that if a function ƒ : C n → C is analytic at λ( a), where a ϵ C m , and is symmetric with respect to a certain partition induced by λ( a), then the composite function g(z) = ƒ(λ 1(z),…,λ n(z)) is analytic at a. When z is real, A( z) is symmetric or Hermitian, and the aforementioned assumptions hold, so that g( z) is analytic at a, we also derive formulae for its first and second order partial derivatives. We apply the results to several problems involving eigenvalues.