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

Abstract

The set (or family) of 2-by-3 matrix pencils $A - \lambda B$ comprises 18 structurally different Kronecker structures (canonical forms). The algebraic and geometric characteristics of the generic and the 17 nongeneric cases are examined in full detail. The complete closure hierarchy of the orbits of all different Kronecker structures is derived and presented in a closure graph that shows 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 shows 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 $(\delta A,\delta B)$ of a generic 2-by-3 pencil $A - \lambda B$ such that $( A + \delta A ) - \lambda ( B + \delta B )$ has a specific nongeneric Kronecker structure are presented. First, explicit expressions for the perturbations that transfer $A - \lambda B$ to a specified nongeneric form are derived. In this context tractable and intractable perturbations are defined. Second, a modified GUPTRI that computes a specified Kronecker structure of a generic pencil is used. Perturbations devised to impose a certain nongeneric structure are computed in a way that guarantees one will find a Kronecker canonical form (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 nongeneric structures under random perturbations in finite 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.