Title: "Differentiability Properties of the Eigenvalues of Real Symmetric Matrices (and Related Topics)."

Abstract: Functions of the eigenvalues of symmetric matrices find applications in many areas of mathematics such as linear algebra, optimization, random matrix theory, multivariate statistical analysis, etc. An important question that arises is to measure how the function value changes when the symmetric matrix is perturbed. The magnitude of the change is connected to the degree of smoothness of the eigenvalues and the function.

I will begin with an overview of related differentiability results and continue with original results characterizing completely when a {\it spectral} function is twice differentiable.

A function $F$ on the space of $n$-by-$n$ real symmetric matrices is called {\it spectral} if it depends only on the eigenvalues of its argument. Spectral functions are just symmetric functions of the eigenvalues. I will show that a spectral function is twice (continuously) differentiable at a matrix if and only if the corresponding symmetric function is twice (continuously) differentiable at the vector of eigenvalues. I will give a concise and easy to use formula for the Hessian of the spectral function.