A standard way of treating the polynomial eigenvalue problem P (λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P ), and their intersection DL(P ), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which...