Partial differential equations for Eigenvalues: sensitivity and perturbation analysis

Abstract

The well-known Wilkinson expressions for the first derivatives of (ordinary) eigen-values and eigenvectors of simple matrices, in terms of the set of eigenvalues and eigenvectors, are redifferentiated and combined to obtain partial differential equations for the eigenvalues. Analogous expressions are obtained for the first derivatives of generalised eigenvalues and eigenvectors of simple pairs of matrices (A, B) , defined by . Again, redifferentiation and combination yields slightly more complicated partial differential equations for the generalised eigenvalues. When the matrices depend on a few parameters θ 1 , θ 2 , …, the resulting differential equations for the eigenvalues, with those parameters as independent variables, can easily be derived. These parametric equations are explicit representations of analytic perturbation results of Kato, expressed by him as rather abstract complex matrix integrals. Connections with bounds for eigenvalues derived by Stewart and Sun can also be made. Two applications are exhibited, the first being to a broken symmetry problem, the second being to working out the second-order perturbations for a classical problem in the theory of waves in cold plasmas.