On Pseudospectra, Critical Points, and Multiple Eigenvalues of Matrix Pencils

We develop a general framework for defining and analyzing pseudospectra of matrix pencils. The framework so developed unifies various definitions of pseudospectra of matrix pencils proposed in the literature. We introduce and analyze critical points of backward errors of approximate eigenvalues of matrix pencils and show that each critical point is a multiple eigenvalue of an appropriately perturbed pencil. We show that common boundary points of components of pseudospectra of a matrix pencil are critical points. In particular, we show that a minimal critical point can be read off from the pseudospectra of matrix pencils. Hence we show that a solution of Wilkinson's problem for a matrix pencil can be read off from the pseudospectra of the matrix pencil.