Differentiating Generalized Eigenvalues and Eigenvectors

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.