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]. Key words: matrix polynomial, matrix pencil, linearization, strong linearization, shifted sum, companion form AMS subject classification. 65F15, 15A18, 15A22.

The standard way of solving the polynomial eigenvalue problem of degree m in n × n matrices is to “linearize” to a pencil in mn × mn matrices and solve the generalized eigenvalue problem. For a given polynomial, P, infinitely many linearizations exist and they can have widely varying eigenvalue condition numbers. We investigate the conditioning of linearizations from a vector space DL(P) of pencils recently identified and studied by Mackey, Mackey, Mehl, and Mehrmann. We look for the best conditioned linearization and compare the conditioning with that of the original polynomial. Two particular pencils are shown always to be almost optimal over linearizations in DL(P) for eigenvalues of modulus greater than or less than 1, respectively, provided that the problem is not too badly scaled and that the pencils are linearizations. Moreover, under this scaling assumption, these pencils are shown to be about as well conditioned as the original polynomial. For quadratic eigenvalue problems that are not too heavily damped, a simple scaling is shown to convert the problem to one that is well scaled. We also analyze the eigenvalue conditioning of the widely used first and second companion linearizations. The conditioning of the first companion linearization relative to that of P is shown to depend on the coefficient matrix norms, the eigenvalue, and the left eigenvectors of the linearization and of P. The companion form is found to be potentially much

Abstract. The most widely used approach for solving the polynomial eigenvalue problem P(λ)x = ��m i=0 λi � Ai x =0inn × n matrices Ai is to linearize to produce a larger order pencil L(λ) =λX + Y, whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial P, infinitely many linearizations L exist and approximate eigenpairs of P computed via linearization can have widely varying backward errors. We show that if a certain one-sided factorization relating L to P can be found then a simple formula permits recovery of right eigenvectors of P from those of L, and the backward error of an approximate eigenpair of P can be bounded in terms of the backward error for the corresponding approximate eigenpair of L. A similar factorization has the same implications for left eigenvectors. We use this technique to derive backward error bounds depending only on the norms of the Ai for the companion pencils and for the vector space DL(P) of pencils recently identified by Mackey, Mackey, Mehl, and Mehrmann. In all cases, sufficient conditions are identified for an optimal backward error for P. These results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of P. Other contributions of this work are a block scaling of the companion pencils

Abstract. Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomial (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of linearizations that 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 not found

Abstract. Hyperbolic matrix polynomials are an important class of Hermitian matrix polynomials that contain overdamped quadratics as a special case. They share with definite pencils the spectral property that their eigenvalues are real and semisimple. We extend the definition of hyperbolic matrix polynomial in a way that relaxes the requirement of definiteness of the leading coefficient matrix, yielding what we call definite polynomials. We show that this class of polynomials has an elegant characterization in terms of definiteness intervals on the extended real line, and that it includes definite pencils as a special case. A fundamental question is whether a definite matrix polynomial P can be linearized in a structure-preserving way. We show that the answer to this question is affirmative: P is definite if and only if it has a definite linearization in H(P), a certain vector space of Hermitian pencils; and for definite P we give a complete characterization of all the linearizations in H(P) that are definite. For the important special case of quadratics, we show how a definite quadratic polynomial can be transformed into a definite linearization with a positive definite leading coefficient matrix—a form that is particularly attractive numerically.

Abstract. Rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical applications. In this paper, we propose a linearization-based method to solve the rational eigenvalue problem. The proposed method converts the rational eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploits and preserves the structure and properties of the original rational eigenvalue problem. For example, the low-rank property leads to a trimmed linearization. We show that solving a class of rational eigenvalue problems is just as convenient and efficient as solving linear eigenvalue problems. Key words. Rational eigenvalue problem, linearization, nonlinear eigenvalue problem AMS subject classifications. 65F15, 65F50, 15A18

The spectral properties of Hermitian matrix polynomials with real eigenvalues have been extensively studied, through classes such as the definite or definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix polynomials, and overdamped or gyroscopically stabilized quadratics. We give a unified treatment of these and related classes that uses the eigenvalue type (or sign characteristic) as a common thread. Equivalent conditions are given for each class in a consistent format. We show that these classes form a hierarchy, all of which are contained in the new class of quasidefinite matrix polynomials. As well as collecting and unifying existing results, we make several new contributions. We propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line. By analyzing their effect on eigenvalue type, we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient, which is important for treating definite and quasidefinite polynomials. We also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable

Abstract not found

We have not known a single great scientist who could not discourse freely and interestingly with a child. Can it be that the haters of clarity have nothing to say, have observed noth-ing, have no clear picture of even their own fields? A dull man seems to be a dull man no matter what his field, and of course it is the right of a dull scientest to protect himself with feathers and robes, emblems and degrees, as do other dull men who are potentates and grand imperial rulers of lodges of dull men. John Steinbeck, the Log From the Sea of Cortez. To my parents and grandparents iv Preface This thesis concerns the quantities indicating the degree of robustness of a dy-namical system. Apart from the intuitive motivation that a dynamical system is surrounded by uncertainties and we would like to know how much uncer-