Variational Principles for Eigenvalues of Nonsymmetric Matrices
Giles Auchmuty · SIAM Journal on Matrix Analysis and Applications · 1989
Some novel variational principles for finding real and complex eigenvalues of the generalized eigenproblem $Ax = \lambda Dx$ are formulated and analyzed. A is a general matrix, D is assumed to be real symmetric and positive definite. One class of principles is based on constrained minimization on the set where $\| Dx \| = 1$. The other class involves the minimization of smooth functions on the complement of certain closed convex sets. The minima of these functions occur either at certain eigenvectors, or certain singular vectors, of the problem. The eigenvalue is determined as a certain functional of the minimizes. A numerical implementation of these principles is described.