# Structured Linearizations for Matrix Polynomials

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## Abstract

The classical approach to investigating polynomial eigenvalue problems is linearization, where the underlying matrix 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, denoted DL(P), 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. Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial; perhaps the most well-known are symmetric and Hermitian polynomials. In this thesis we also identify several less well-known types of structured polynomial (e.g., palindromic, even, odd), explore the relationships between them, and illustrate their appearance in a variety of applications. Special classes of linearizations that respect the structure of these polynomials, and therefore preserve symmetries in their spectra, are introduced and investigated. We analyze the existence and uniqueness of such linearizations, and show how they may be systematically constructed. The infinitely many linearizations of any given polynomial P can have widely varying eigenvalue condition numbers. We investigate the conditioning of linearizations from DL(P), looking for the best conditioned linearization in that space and comparing its conditioning with that of the original polynomial. We also analyze the eigenvalue conditioning of the widely used first and second companion linearizations, and find that they can potentially be much more ill conditioned than P. Our results are phrased in terms of both the standard relative condition number and the condition number of Dedieu and Tisseur for the problem in homogeneous form, this latter condition number having the advantage of applying to zero and infinite eigenvalues.

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