Eigenvalue characterization of some structured matrix pencils under linear perturbation

Abstract

We study the effect of linear perturbations on three families of matrix pencils. The matrix pairs of the first two families are Hermitian/skew-Hermitian with special $3\times 3$ block cases appeared in continuous-time control, and the matrix pairs of the third family are special $3\times 3$ non-Hermitian block matrices appeared in discrete-time control. For the first family of matrix pencils and more general cases of the second family of matrix pencils, based on the properties of the involved matrices, we obtain some upper or lower bounds on the set of eigenvalues of linearly perturbed matrix pencils which are on the imaginary axis. Studying a special $3\times 3$ block matrix pencil, which is associated with continuous-time control, leads us to some linear perturbation that do not preserve (properly) the structure of the matrices. This, in turn, leads to a numerical technique for finding the nearest Hermitian/skew-Hermitian matrix pencil which can satisfy conditions such that, for some nonzero real perturbation parameter, some or all of its eigenvalues lie on the imaginary axis. We also study the linearly perturbed matrix pencils, associated with discrete-time control, using an one-to-one equivalence between the matrix pencil of continuous-time problem and the matrix pencil of discrete-time problem.