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Our purpose is to give new proofs of several known results aboutperturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closureof orbit of a Kronecker canonical pencil $A-\lambda B$ in terms of inequalitieswith pencil invariants. In more detail, Pokrzywa described all Kroneckercanonical pencils $K-\lambda L$ such that each neighborhood of $A-\lambda B$contains a pencil whose Kronecker canonical form is $K-\lambda L$. Anothersolution of this problem was given by Klaus Bongartz (1996) by methods ofrepresentation theory.  We give a direct and constructive proof of Pokrzywa's theorem. We reduce itsproof to the cases of matrices under similarity and of matrix pencils$P-\lambda Q$ that are direct sums of two indecomposable Kronecker canonicalpencils. We calculate the Kronecker forms of all pencils in a neighborhood ofsuch a pencil $P-\lambda Q$. In fact, we calculate the Kronecker forms of onlythose pencils that belong to a miniversal deformation of $P-\lambda Q$, whichis sufficient since all pencils in a neighborhood of $P-\lambda Q$ are reducedto them by smooth strict equivalence transformations.

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