On the dimension of orbits of matrix pencils under strict equivalence

De Terán, Fernando, Dopico, Froilán M., Pagacz, Patryk

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.