Till Örebro universitet

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.