Analysis of structured polynomial eigenvalue problems (original) (raw)

Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification

Linear Algebra and its Applications, 2012

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 by structure preserving congruence, and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial, thereby identifying an important new class of diagonalizable matrix polynomials.

An algebraic theory about semiclassical and classical matrix orthogonal polynomials

In this paper we introduce an algebraic theory of classical matrix orthogonal polynomials as a particular case of the semi-classical ones, defined by a distributional equation for the corresponding orthogonality functional. This leads to several properties that characterize the classical matrix families, among them, a structure relation and a second order differo-differential equation. In the particular case of Hermite type matrix polynomials we obtain all the parameters associated with the family and we prove that they satisfy, not only a differo-differential equation, but a second order differential one, as it can be seen in the scalar case.

On Matrix Polynomials with Real Roots

SIAM Journal on Matrix Analysis and Applications, 2005

It is proved that the roots of combinations of matrix polynomials with real roots can be recast as eigenvalues of combinations of real symmetric matrices, under certain hypotheses. The proof is based on recent solution of the Lax conjecture. Several applications and corollaries, in particular concerning hyperbolic matrix polynomials, are presented.

On pseudo Hermite matrix polynomials of two variables

Banach Journal of Mathematical Analysis, 2010

The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix polynomials are underlined.

A Contribution To The Polynomial Eigen Problem

2014

The relationship between eigenstructure (eigenvalues<br> and eigenvectors) and latent structure (latent roots and latent vectors)<br> is established. In control theory eigenstructure is associated with<br> the state space description of a dynamic multi-variable system and<br> a latent structure is associated with its matrix fraction description.<br> Beginning with block controller and block observer state space forms<br> and moving on to any general state space form, we develop the<br> identities that relate eigenvectors and latent vectors in either direction.<br> Numerical examples illustrate this result. A brief discussion of the<br> potential of these identities in linear control system design follows.<br> Additionally, we present a consequent result: a quick and easy<br> method to solve the polynomial eigenvalue problem for regular matrix<br> polynomials.

Associated polynomials, spectral matrices and the bispectral problem

Methods and Applications of Analysis, 1999

The associated Hermite, Laguerre, Jacobi, and Bessel polynomials appear naturally when Bochner's problem [5] of characterizing orthogonal polynomials satisfying a second-order differential equation is extended to doubly infinite tridiagonal matrices. We obtain an explicit expression for the spectral matrix measure corresponding to the associated doubly infinite tridiagonal matrix in the Jacobi case. We show that, in an appropriate basis of "bispectral" functions, the spectral matrix can be put into a nice diagonal form, restoring the simplicity of the familiar orthogonality relations satisfied by the Jacobi polynomials.

LINEARIZATIONS OF POLYNOMIAL MATRICES WITH SYMMETRIES AND THEIR APPLICATIONS

2005

In an earlier paper by the present authors, a new family of companion forms associated with a regular polynomial matrix was presented, generalizing similar results by M. Fiedler who considered the scalar case. This family of companion forms preserves both the finite and infinite elementary divisor structure of the original polynomial matrix, thus all its members can be seen as linearizations of the corresponding polynomial matrix. In this note, its applications on polynomial matrices with symmetries, which appear in a number of engineering fields, are examined.

Diagonalization of quadratic matrix polynomials

Systems & Control Letters, 2010

Solving the quadratic eigenvalue problem is critical in several applications in control and systems theory. One alternative to solve this problem is to reduce the matrix to a diagonal form so that its eigenvalue structure can be recognized in the diagonal of the equivalent matrix. There are two major categories of diagonalizable systems. The first category concerns systems that are strictly equivalent. The second category is much wider and consists of systems for which their linearizations are strictly equivalent. Here we are concerned with methods to reduce the linearization of a quadratic matrix polynomial to a diagonal form. We give necessary and sufficient conditions for a system to have a diagonalization and we argue on two different methods to diagonalize a system (via its linearization) that one can find in the literature. Based on the results presented here, we conclude that the problem is still open.