Differentiating Generalized Eigenvalues and Eigenvectors

TL;DR

This paper derives formulas for the derivatives of generalized eigenvalues and eigenvectors of symmetric matrices, providing computational tools and applications in multivariate data analysis.

Contribution

It introduces explicit formulas for derivatives of generalized eigenvalues/eigenvectors and singular values/vectors, including R implementations and validation methods.

Findings

Derived first and second derivative formulas for generalized eigenvalues/eigenvectors.

Provided R functions for computing these derivatives.

Validated formulas against numerical differentiation.

Abstract

We give formulae for first and second derivatives of generalized eigenvalues/eigenvectors of symmetric matrices and generalized singular values/singular vectors of rectangular matrices when the matrices are linear or nonlinear functions of a vector of parameters. In addition we provide functions in R to compute these derivatives, both in the general case and in various special cases. Formulae are checked against Jacobians and Hessians computed by numerical differentiation. Some applications to multivariate data analysis are discussed.