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-lambda B comprises 18 s tructurally different Kronecker structures (canonical forms). The alge braic and geometric characteristics of the generic and the 17 nongener ic cases are examined in full detail. The complete closure hierarchy o f the orbits of all different Kronecker structures is derived and pres ented in a closure graph that shows how the structures relate to each other in the la-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 pre sented in a labeled closure graph. The node and are labels shows geome tric characteristics of an orbit's Kronecker structure and the change of geometric characteristics when transiting to an adjacent node, resp ectively. Computable normwise bounds for the smallest perturbations (d elta A, delta B) of a generic 2-by-3 pencil A lambda B such that (A+de lta 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 th is 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 non generic structure are computed in a way that guarantees one will find a Kronecker canonical form (KCF) on the closure of the orbit of the in tended KCF. Both approaches are illustrated by computational experimen ts. Moreover, a study of the behaviour of the nongeneric structures un der random perturbations in finite precision arithmetic (using the GUP TRI software) show for which sizes of perturbations the structures are invariant and also that structure transitions occur in accordance wit h the closure hierarchy. Finally, some of the results are extended to the general m-by-(m+1) case.