Abstract. 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 [7], [8], [12].

Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomials (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of linearizations which reflect the structure of these polynomials, and therefore preserve symmetries in their spectra, is introduced and investigated. We analyze the existence and uniqueness of such linearizations and show how they may be systematically constructed.

Abstract.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 L 1 (P ) and L 2 (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 their constituent pencils inherit structure from P . For arbitrary polynomials we show that every pencil in DL(P ) is block symmetric and we obtain a convenient basis for DL(P ) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P ) pencils in a sequence constructed by Lancaster in the 1960s generate DL(P ). When P is symmetric, we show that the symmetric pencils in L 1 (P ) comprise DL(P ), while for Hermitian P the Hermitian pencils in L 1 (P ) form a proper subset of DL(P ) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a self-contained treatment of some of the key properties of DL(P ) together with some new, more concise proofs.

We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated.

Abstract. Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palindromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular and singular T -palindromic matrix polynomials over arbitrary fields. The invariant polynomials are shown to inherit palindromicity, and their structure is described in detail. Jordan structures of palindromic matrix polynomials are characterized, and necessary conditions for the existence of structured linearizations established. In the odd degree case, a constructive procedure for building palindromic linearizations shows that the necessary conditions are sufficient as well. The Smith form for * -palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of characteristic two are presented.