To Örebro University

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.

Canonical forms provide a fundamental framework for studying matrices and matrix pencils through invariants under various equivalence transformations. These forms reveal structural information about the underlying objects, such as eigenvalues, their multiplicities, and minimal indices. This information plays an important role in differential-algebraic equations, control theory, generalized eigenvalue problems, and related areas. However, in practical applications, canonical invariants are often sensitive to perturbations arising from measurement errors and numerical computations. Closely related to perturbation theory is the notion of the distance to singularity. This distance quantifies how far a matrix pencil is from becoming singular. This thesis investigates canonical forms for pairs of matrices associated with linear time-invariant dissipative Hamiltonian systems under suitable equivalence transformations. Furthermore, it studies minimal changes in the eigenstructure of skew-symmetric matrix pencils under small perturbations by developing theories related to orbits and bundles of skew-symmetric matrix pencils under congruence. Finally, it presents an algorithm for computing a nearby singular matrix pencil from a given regular matrix pencil, where the distance between the two is measured in the Frobenius norm. The method builds on recent advances in the theory of matrix pencil factorizations. The result is then extended to matrix polynomials.

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.