Generalized Hadamard Product and the Derivatives of Spectral Functions
Hristo S. Sendov · SIAM Journal on Matrix Analysis and Applications · 2006
Real valued functions, $F(X)$, on a symmetric matrix argument are called spectral if $F(U^TXU) = F(X)$ for every orthogonal matrix U and $X \in \mathrm{dom\,} F$. We are interested in a description of the higher order derivatives (when they exist) of F with respect to X. Formulae for the gradient and the Hessian of F are given in [A. S. Lewis, Math. Oper. Res., 21 (1996), pp. 576–588] and [A. S. Lewis and H. S. Sendov, SIAM Matrix Anal. Appl., 23 (2001), pp. 368–386]. In this work we present common features of these two formulae that are preserved in the higher order derivatives.