Derivatives of Eigenvalues and Eigenvectors of Matrix Functions

Analysis is made of the existence of derivatives, the effect of normalization strategies, and the solvability and condition of the bordered matrix equations arising naturally in this context.

Abstract

For an $n \times n$ matrix-valued function $L( {\boldsymbol \rho} ,\lambda )$, where ${\boldsymbol \rho} $ is a vector of independent parameters and $\lambda$ is an eigenparameter, the eigenvalue-eigenvector problem has the form $L ( {\boldsymbol \rho} ,\lambda ({\boldsymbol \rho} ){\bf x} ( {\boldsymbol \rho} ) ) = {\bf 0}$. Real or complex values for ${\boldsymbol \rho}$ and $\lambda$ are admitted, and L is assumed to depend analytically on these variables. In particular, nonlinear dependence on $\lambda$ is the main concern. On the assumption that the eigenvalue-eigenvector problem itself can be satisfactorily solved, a study is made of the derivatives (sensitivities) of $\lambda$ and ${\bf x}$ with respect to ${\boldsymbol \rho}$. Analysis is made of the existence of derivatives, the effect of normalization strategies, and the solvability and condition of the bordered matrix equations arising naturally in this context. Implications for various classical eigenvalue problems $L({\boldsymbol \rho} ,\lambda ) = A( {\boldsymbol \rho} ) - \lambda I$ are clarified.