On the Higher-Order Derivatives of Spectral Functions: Two Special Cases

TL;DR

This paper develops streamlined formulas for higher-order derivatives of spectral functions of symmetric matrices, focusing on two special cases, using tensor calculus and generalized Hadamard products to simplify the differentiation process.

Contribution

It provides new, concise formulas for derivatives of spectral functions in two specific cases, improving upon previous methods with a tensor-based approach.

Findings

Derived formulas for derivatives of eigenvalue functions with distinct eigenvalues

Established derivatives for separable symmetric functions at arbitrary matrices

Re-derived the Hessian formula for general spectral functions

Abstract

In this work we use the tensorial language developed in [8] and [9] to differentiate functions of eigenvalues of symmetric matrices. We describe the formulae for the k-th derivative of such functions in two cases. The first case concerns the derivatives of the composition of an arbitrary differentiable function with the eigenvalues at a matrix with distinct eigenvalues. The second development describes the derivatives of the composition of a separable symmetric function with the eigenvalues at an arbitrary symmetric matrix. In the concluding section we re-derive the formula for the Hessian of a general spectral function at an arbitrary point. Our approach leads to a shorter, streamlined derivation than the original in [6]. The language we use, based on the generalized Hadamard product, allows us to view the differentiation of spectral functions as a routine calculus-type procedure.