DISTANCE PROBLEMS FOR HERMITIAN MATRIX PENCILS WITH EIGENVALUES OF DEFINITE TYPE

Given a Hermitian matrix pencil L(z) = zA - B with only real eigenvalues that are either of positive or negative type, the distance to a nearest Hermitian pencil outside the class is considered with respect to a specified norm. These problems are considered in the setting of the Hermitian epsilon-pseudospectra of L(z) with a proposed homogeneous form of the definition of eigenvalue type playing an important role in the investigations. A significant outcome of this analysis is a bisection-type algorithm for computing the Crawford number of a definite pencil and a nearest Hermitian pencil that is not definite. Each step of the algorithm requires the computation of the smallest eigenvalue(s) of a positive definite matrix of the same size as the original definite pencil and corresponding eigenvector(s). The results may also be extended to compute the distance from a definitizable pencil to a nearest Hermitian pencil that is not definitizable and a smallest Hermitian perturbation that attains this distance. The ideas extend to computing solutions for similar distance problems involving skew-Hermitian, *-even, and *-odd pencils.