Companion form linearizations do not reflect any structure that may be present in the original polynomial , so that their use in numerics may be problematic. Another issue is conditioning, and numerics may produce different results for each linearization. A further issue for linearizations concerns eigenvalues at infinity, as they occur in circuit simulation and wave guides.
The authors show how to systematically generate, for any regular matrix polynomial, large classes of linearizations that address those issues. This is done by using a natural generalization of companion forms, resulting in two vector spaces of pencils, with an elegant characterization of exactly which pencils in the intersection of the two spaces are linearizations for . The eigenvectors of any such pencil are rather simply related to those of .