Quadratic expansions of spectral functions

A function, F, on the space of n x n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F(A) = F(UAU(T)) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on RI: those that are invariant under arbitrary swapping of their arguments. In this paper, we show that a spectral function has a quadratic expansion around a point A if and only if its corresponding symmetric function has quadratic expansion around lambda (A) (the vector of eigenvalues). We also give a concise and easy to use formula for the 'Hessian' of the spectral function, In the case of convex functions we show that a positive definite 'Hessian' of f implies positive definiteness of the 'Hessian' of F. (C) 2002 Elsevier Science Inc. All rights reserved.