Vector Spaces of Linearizations for Matrix Polynomials

The classical approach to investigating polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes be required. Given a matrix polynomial $P$, we develop a systematic approach to generating large classes of linearizations for $P$. We show how to simply construct two vector spaces of pencils that generalize the companion forms of $P$, and prove that almost all of these pencils are linearizations for $P$. Eigenvectors of these pencils are shown to be closely related to those of $P$. A distinguished subspace is then isolated, and the special properties of these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations.