Perturbation theory of matrix pencils through miniversal deformations

We give new proofs of several known results about perturbations of matrix pencils. In particular, we give a direct and constructive proof of Andrzej Pokrzywa's theorem (1983), in which the closure of the orbit of each Kronecker canonical matrix pencil is described in terms of inequalities for invariants of matrix pencils. A more abstract description is given by Klaus Bongartz (1996) by methods of representation theory. We formulate and prove Pokrzywa's theorem in terms of successive replacements of direct summands in a Kronecker canonical pencil. First we show that it is sufficient to prove Pokrzywa's theorem in two cases: for matrices under similarity and for each matrix pencil P − λ Q that is a direct sum of two indecomposable pencils. Then we calculate the Kronecker canonical form of pencils that are close to P − λ Q . In fact, the Kronecker canonical form is calculated for only those pencils that belong to a miniversal deformation of P − λ Q . This is sufficient since all pencils in a neighborhood of P − λ Q are reduced to them by a smooth strict equivalence transformation.