On the dimension of orbits of matrix pencils under strict equivalence

Abstract

We prove that, given two matrix pencils L and M, if M belongs to the closure of the orbit of L under strict equivalence, then the dimension of the orbit of M is smaller than or equal to the dimension of the orbit of M, and the equality is only attained when M belongs to the orbit of L. Our proof uses only the majorization involving the eigenstructures of L and M 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.The authors thank Andrii Dmytryshyn for suggesting the use of Theorem 6 to prove the main result. This work is part of grant PID2023-147366NB-I00 funded by MICIU/AEI/ 10.13039/501100011033 and FEDER/UE. Also funded by RED2022-134176-T and by the program Excellence Initiative - Research University at the Jagiellonian University in Kraków