The Set of 2-by-3 Matrix Pencils - Kronecker Structures and Their Transitions under Perturbations

The set (or family) of 2-by-3 matrix pencils A ? B comprises 18 structurally diierent Kronecker structures (canonical forms). The algebraic and geometric characteristics of the generic and the 17 non-generic cases are examined in full detail. The complete closure hierarchy of the orbits of all diierent Kronecker structures is derived and presented in a closure graph that show how the structures relate to each other in the 12-dimensional space spanned by the set of 2-by-3 pencils. Necessary conditions on perturbations for transiting from the orbit of one Kronecker structure to another in the closure hierarchy are presented in a labeled closure graph. The node and arc labels show geometric characteristics of an orbit's Kronecker structure and the change of geometric characteristics when transiting to an adjacent node, respectively. Computable normwise bounds for the smallest perturbations (A; B) of a generic 2-by-3 pencil A?B such that (A+A)? (B +B) has a speciic non-generic Kronecker structure are presented. First explicit expressions for the perturbations that transfer A ? B to a speciied non-generic form are derived. In this context tractable and intractable perturbations are deened. Secondly, a modiied GUPTRI that computes a speciied Kronecker structure of a generic pencil is used. Perturbations devised to impose a certain non-generic structure is computed in a way that guarantees to nd a KCF on the closure of the orbit of the intended KCF. Both approaches are illustrated by computational experiments. Moreover, a study of the behaviour of the non-generic structures under random perturbations in nite precision arithmetic (using the GUPTRI software) show for which sizes of perturbations the structures are invariant and also that structure transitions occur in accordance with the closure hierarchy. Finally, some of the results are extended to the general m-by-(m + 1) case.