Partial Derivatives of Repeated Eigenvalues and Their Eigenvectors

The analysis of inverse problems in linear modeling often require the sensitivities of the eigenvalues and eigenvectors. The calculation of these sensitivities is mathematically related to the corresponding partial derivatives, which do not exist for any parameterization. Inasmuch as eigenvalues and eigenvectors are coupled by the constitutional equation of the general eigenvalue problem, their derivatives are coupled, too. Conditions on the parameterization are derived and formulated as theorems, which ensure the existence of the partial derivatives of the eigenvalues and eigenvectors with respect to these parameters. The application of the theorems is demonstrated by examples.