On the dimension of orbits of matrix pencils under strict equivalence

TL;DR

This paper establishes a relationship between the orbit dimensions of matrix pencils under strict equivalence, showing that orbit closure inclusion implies a non-increasing dimension, with equality only when the pencils are strictly equivalent.

Contribution

It provides a new proof linking orbit closure and dimension inequalities using eigenstructure majorization and codimension formulas.

Findings

Orbit dimension decreases under closure inclusion.

Equality in dimension occurs only for strictly equivalent pencils.

The proof relies on eigenstructure majorization and codimension formulas.

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 $L$, 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.