🤖 AI Summary
Efficient and accurate computation of higher-order derivatives of generalized eigenvalues/vectors for symmetric matrices and generalized singular values/vectors for rectangular matrices—parameterized linearly or nonlinearly—is essential yet theoretically underdeveloped, especially under degeneracy and practical constraints.
Method: We rigorously derive closed-form expressions for first- and second-order analytical derivatives using matrix differential calculus, validate Jacobian and Hessian matrices via numerical differentiation, and implement an open-source R package supporting diverse degenerate cases and application-specific constraints.
Contribution/Results: This work establishes the first unified, complete high-order differential theory framework for these spectral decompositions. The proposed method substantially improves both computational efficiency and numerical accuracy of derivative evaluation. It has been successfully applied to multivariate statistical modeling and covariance structure optimization, providing a rigorous theoretical foundation and practical computational infrastructure for parameter sensitivity analysis and gradient-based optimization in eigen-decomposition–dependent models.
📝 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.