Quadratic expansions of spectral functions Original Research Article

A function, F, on the space of n×n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F(A)=F(UAUT) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on image : 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 λ(A) (the vector of eigenvalues). We also give a concise and easy to use formula for the ‘Hessianʹ of the spectral function. In the case of convex functions we show that a positive definite ‘Hessianʹ of f implies positive definiteness of the ‘Hessianʹ of F