Regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings

Two m×n pencils A+λB and A′+λB′ over F=R or C are said to be topologically equivalent if the pairs of linear mappings A,B:Fn→Fm and A′,B′:Fn→Fm coincide up to homeomorphisms of the spaces Fn and Fm. We prove that two pencils are topologically equivalent if and only if their regularizing decompositions coincide up to permutation of summands and replacement of D by a nonsingular matrix D′ such that the linear operators D,D′:Fr→Fr coincide up to a homeomorphism of Fr.