Linearizations of Hermitian Matrix Polynomials Preserving the Sign Characteristic

The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of a given Hermitian matrix polynomial and identify several families of such linearizations that can be constructed very easily from the coefficients of the polynomial

Analysis of structured polynomial eigenvalue problems

This thesis considers Hermitian/symmetric, alternating and palindromic matrix polynomials which all arise frequently in a variety of applications, such as vibration analysis of dynamical systems and optimal control problems. A classification of Hermitian matrix polynomials whose eigenvalues belong to the extended real line, with each eigenvalue being of definite type, is provided first. We call such polynomials quasidefinite. Definite pencils, definitizable pencils, overdamped quadratics, gyroscopically stabilized quadratics, (quasi)hyperbolic and definite matrix polynomials are all quasidefinite. We show, using homogeneous rotations, special Hermitian linearizations and a new characterization of hyperbolic matrix polynomials, that the main common thread between these many subclasses is the distribution of their eigenvalue types. We also identify, amongst all quasihyperbolic matrix polynomials, those that can be diagonalized by a congruence transformation applied to a Hermitian linearization of the matrix polynomial while maintaining the structure of the linearization. Secondly, we generalize the notion of self-adjoint standard triples associated with Hermitian matrix polynomials in Gohberg, Lancaster and Rodman's theory of matrix polynomials to present spectral decompositions of structured matrix polynomials in terms of standard pairs (X,T), which are either real or complex, plus a parameter matrix S that acquires particular properties depending on the structure under investigation. These decompositions are mainly an extension of the Jordan canonical form for a matrix over the real or complex field so we investigate the important special case of structured Jordan triples. Finally, we use the concept of structured Jordan triples to solve a structured inverse polynomial eigenvalue problem. As a consequence, we can enlarge the collection of nonlinear eigenvalue problems [NLEVP, 2010] by generating quadratic and cubic quasidefinite matrix polynomials in different subclasses from some given spectral data by solving an appropriate inverse eigenvalue problem. For the quadratic case, we employ available algorithms to provide tridiagonal definite matrix polynomials.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Large vector spaces of block-symmetric strong linearizations of matrix polynomials

Given a matrix polynomial P(lambda) = Sigma(k)(i=0) lambda(i) A(i) of degree k, where A(i) are n x n matrices with entries in a field F, the development of linearizations of P(lambda) that preserve whatever structure P(lambda) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P(lambda) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P(lambda) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space of dimension k of block-symmetric pencils, called DL(P), such that most of its pencils are linearizations. One drawback of the pencils in DL(P) is that none of them is a linearization when P(lambda) is singular. In this paper we introduce new vector spaces of block,symmetric pencils, most of which are strong linearizations of P(lambda). The dimensions of these spaces are O(n(2)), which, for n >= root k, are much larger than the dimension of DL(P). When k is odd, many of these vector spaces contain linearizations also when P(lambda) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k x k block-matrices whose n x n blocks are of the form 0, +/-alpha I-n, +/-alpha A(i), or arbitrary n x n matrices, where a is an arbitrary nonzero scalar.The research of F. M. Dopico was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2012-3254

Diagonalization of polynomial matrices

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: M. Eulàlia Montoro López[en] This work studies the diagonalization of second degree polynomial matrices. First, all the concepts needed to understand the theory on this type of matrix are defined. Then, the most important working tools for solving the problem are introduced: the Smith canonical form and the linearization of polynomial matrices. Finally, it is deduced for which 2nd degree matrices there is a diagonalization

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behavior of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients

On the Rellich eigendecomposition of para-Hermitian matrices and the sign characteristics of ∗*-palindromic matrix polynomials

We study the eigendecompositions of para-Hermitian matrices $H(z)$, that is, matrix-valued functions that are analytic and Hermitian on the unit circle $S^1 \subset \mathbb C$. In particular, we fill existing gaps in the literature and prove the existence of a decomposition $H(z)=U(z)D(z)U(z)^P$ where, for all $z \in S^1$, $U(z)$ is unitary, $U(z)^P=U(z)^*$ is its conjugate transpose, and $D(z)$ is real diagonal; moreover, $U(z)$ and $D(z)$ are analytic functions of $w=z^{1/N}$ for some positive integer $N$, and $U(z)^P$ is the so-called para-Hermitian conjugate of $U(z)$. This generalizes the celebrated theorem of Rellich for matrix-valued functions that are analytic and Hermitian on the real line. We also show that there also exists a decomposition $H(z)=V(z)C(z)V(z)^P$ where $C(z)$ is pseudo-circulant, $V(z)$ is unitary and both are analytic in $z$. We argue that, in fact, a version of Rellich's theorem can be stated for matrix-valued function that are analytic and Hermitian on any line or any circle on the complex plane. Moreover, we extend these results to para-Hermitian matrices whose entries are Puiseux series (that is, on the unit circle they are analytic in $w$ but possibly not in $z$). Finally, we discuss the implications of our results on the singular value decomposition of a matrix whose entries are $S^1$-analytic functions of $w$, and on the sign characteristics associated with unimodular eigenvalues of $*$-palindromic matrix polynomials