Some Observations on the Sensitivity Analysis of Spectral Functions

We examine the formulae for the first and second derivatives of spectral functions of real symmetric matrices to understand why they have the form that they do. The development, which involves only elementary linear algebra and calculus, is based on computing directional derivatives of the spectral function and its gradient along a canonical set of orthonormal matrices. A particular aim of our derivation is an interpretation of the individual terms that appear in the formula given by A. Lewis and H. Sendov for the action of the Hessian of a spectral function. An examination of the formal calculations we use to derive the first derivative formula also leads to a proof of the result on differentiability of spectral functions due to A. Lewis. The directional derivatives we compute also reveal the spectral structure of the Hessian of a spectral function, and can be used to explain the properties of the Newton step for a spectral function.