Derivatives of functions of eigenvalues and eigenvectors for symmetric matrices

This paper discusses the differentiability of a class of functions associated with eigenvalues and eigenvectors of symmetric matrices. Recursive style formulas of partial derivatives for this class of functions are derived and higher order derivatives can be easily obtained from these formulas. Meanwhile, some interesting characteristics of multiple eigenvalues are revealed. Examples involving inverse eigenvalue problems and primary matrix functions are given to illustrate the applications of the results obtained in this paper.