A constructive proof of Pokrzywa's theorem about perturbations of matrix pencils

Abstract

Our purpose is to give new proofs of several known results about perturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closure of orbit of a Kronecker canonical pencil $A-\lambda B$ in terms of inequalities with pencil invariants. In more detail, Pokrzywa described all Kronecker canonical pencils $K-\lambda L$ such that each neighborhood of $A-\lambda B$ contains a pencil whose Kronecker canonical form is $K-\lambda L$. Another solution of this problem was given by Klaus Bongartz (1996) by methods of representation theory. We give a direct and constructive proof of Pokrzywa's theorem. We reduce its proof to the cases of matrices under similarity and of matrix pencils $P-\lambda Q$ that are direct sums of two indecomposable Kronecker canonical pencils. We calculate the Kronecker forms of all pencils in a neighborhood of such a pencil $P-\lambda Q$. In fact, we calculate the Kronecker forms of only those pencils that belong to a miniversal deformation of $P-\lambda Q$, which is sufficient since all pencils in a neighborhood of $P-\lambda Q$ are reduced to them by smooth strict equivalence transformations.