On the dimension of orbits of matrix pencils under strict equivalence

Abstract

We prove that, given two matrix pencils and , if belongs to the closure of the orbit of under strict equivalence, then the dimension of the orbit of is smaller than or equal to the dimension of the orbit of , and the equality is only attained when belongs to the orbit of . Our proof uses only the majorization involving the eigenstructures of and which characterizes the inclusion relationship between orbit closures, together with the formula for the codimension of the orbit of a pencil in terms of its eigenstruture