Title: Generalized Hadamard Product and the Derivatives of Spectral Functions

A function F on the space of n × n 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 every A in its domain. We are interested in the derivatives of these functions with respect to the argument A. Explicit formulae for the gradient and the Hessian of spectral functions are given in [1], [2]. These formulae appear quite different from each other and this obstructs attempts to generalize them. The questions that we will address in this talk are about some common features in these formulae. We propose a language that may aid the description of the higher derivatives of the spectral functions. It is based on a generalization of the Hadamard product between two matrices to a tensor-valued product between k matrices, for k ≥ 1. The form of the formula for the k-th derivative of a spectral function is conjectured.
[1] A.S. LEWIS, Derivatives of spectral functions, Mathematics of Operations Research, 21:576-588, 1996.
[2] A.S. LEWIS and H.S. SENDOV, Twice Differentiable Spectral Functions, SIAM Journal of Matrix Analysis and Applications, 23(2):368-386, 2001.