Second-order directional derivatives of spectral functions

A spectral function of a symmetric matrix X is a function which depends only on the eigenvalues of X, λ1 (X) ≥ λ2(X) ≥≥ λn (X), and may be written as ƒ (λ1 (X), λ2(X), , λn (X)) for some symmetric function ƒ. In this paper, we assume that ƒ is a C1,1 function and discuss second-order directional derivatives of such a spectral function. We obtain an explicit expression of second-order directional derivative for the spectral function.