Till Örebro universitet

The complete eigenstructure, i.e. eigenvalues with multiplicities and minimal indices, of a skew-symmetric matrix pencil may change drastically if the matrix coefficients of the pencil are subjected to (even small) perturbations. These changes can be investigated qualitatively by constructing the stratification (closure hierarchy) graphs of the congruence orbits of the pencils. The results of this paper facilitate the construction of such graphs by providing all closest neighbours for a given node in the graph. More precisely, we prove a necessary and sufficient condition for one congruence orbit of a skew-symmetric matrix pencil, A, to belong to the closure of the congruence orbit of another pencil, B, such that there is no pencil, C, whose orbit contains the closure of the orbit of A and is contained in the closure of the orbit of B.

Small perturbations to the coefficients of a skew-symmetric matrix pencil can cause large changes in its complete eigenstructure, e.g., eigenvalues, their multiplicities, as well as minimal indices may change. In this manuscript, we state three results: (a) the characterization of inclusion between the closures of congruence bundles of skew-symmetric matrix pencils; (b) the necessary and sufficient condition for one congruence bundle of a skew-symmetric matrix pencil, P, to belong to the closure of the congruence bundle of another matrix pencil, Q, such that there is no matrix pencil, R, whose bundle contains the closure of the bundle of P and is contained in the closure of the bundle of Q; and (c) bundles are open in their closures.