Linear Transformations which leave controllable multiinput descriptor systems controllable

AbstractA general linear descriptor system Eitẋ = Ax + Bu, where E, A∈Cn, n, B∈Cn, m x∈Cn, u∈Cm, E singular, is called controllable if rank [αE − βA, B] = n for all (α, β) ≠ (0, 0), α, β∈C. Let f: Cn, n + m → Cn, n + m be a linear transformation form f(X) = CXD. We characterize all such linear transformations that leave the set C = {[αE − βA, B]|rank[αE− βA, B] = n∀(α, β)≠(0, 0), α, β∈C} invariant and show that only the well-known transformations that leave controllability invariant can occur