Triangularizing Quadratic Matrix Polynomials

Quadratic realizability of palindromic matrix polynomials

Given a quadratic two-parameter matrix polynomial Q(λ, μ), we develop a systematic approach to generating a vector space of linear two-parameter matrix polynomials. The purpose for constructing this vector space is that potential linearizations of Q(λ, μ) lie in it. Then, we identify a set of linearizations and describe their constructions. Finally, we determine a class of linearizations for a quadratic two-parameter eigenvalue problem.

We study the eigenvalues and eigenspaces of the quadratic matrix polynomial $M\lambda^2+sD\lambda+K$ as $s\rightarrow\infty$, where $M$ and $K$ are symmetric positive definite and $D$ is symmetric positive semidefinite. This work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically undamped modes such that all nodes in the model vibrate essentially in phase.

Let L be a monic quadratic weakly hyperbolic or hyperbolic n × n matrix polynomial. We solve some direct spectral problems: We prove that the eigenvalues of a compression of L to an (n − 1)-dimensional subspace of ℂn block-interlace and that the eigenvalues of a one-dimensional perturbation of L (−,+)-interlace the eigenvalues of L. We also solve an inverse spectral problem: We identify two given block-interlacing sets of real numbers as the sets of eigenvalues of L and its compression.Mathematics Subject Classification (2000)47A5647A1047A5547A7547B15KeywordsWeakly hyperbolichyperboliccompressionperturbationNevanlinna functioneigenvalueinterlacingroot zonedirect spectral probleminverse spectral problem

It is shown that any regular polynomial matrix over the field of complex numbers that has at most one elementary divisor of degree 3 and whose remaining elementary divisors are of degree no greater than 2 can be factored into linear regular factors.

Low rank perturbations of strongly definitizable transformations and matrix polynomials

Matrix polynomials of the form $P(\lambda)=I\lambda^2+A_1\lambda+A_0$ (where A0 and A1 are n × n Hermitian matrices and $\lambda$ is a complex variable) arise in many applications. The numerical range of such a polynomial is \[ W(P) = \{\lambda \in \mathbb{C} : x^*P(\lambda)x = 0 \mbox{ for some nonzero } x \in \mathbb{C}^n \} \] and it always contains the spectrum of $P(\lambda)$, i.e., the set of zeros of $\textup{det}P(\lambda)$. Properties of the numerical range are developed in detail, taking advantage of the close connection between W(P) and the classical numerical range (field of values) of the (general) complex matrix A:=A0+iA1. Eigenvalues and nondifferentiable points on the boundary are examined and a procedure for the numerical determination of W(P) is presented and used for several illustrations. Some extensions of the theory to more general polynomials $P(\lambda)$ are also discussed, as well as special cases describing vibrating systems.

The matrix polynomial Q(λ) = λI-S(λ), where S(·) is a nonmonic quadratic matrix polynomial with (entrywise) nonnegative square matrix coefficients, will be studied. We describe the distribution of the eigenvalues of Q(·), depending on the sign of function r↦ r - ϱ(S(r)) (here ϱ(·) denotes the spectral radius). The existence of a nonnegative (spectral) matrix root of Q(·) will be related to the existence of a positive r > ϱ(S(r)). Assuming that S(t) is irreducible for one positive t, we describe the spectrum of Q(·) on the circles with radius r for any r = ϱ(S(r)) > 0, and describe the possibilities for the existence of a nonnegative matrix root of Q(·), for the properties of a corresponding M-matrix and the spectral properties of Q(·), depending on the function ϱ(S(·)) and on its derivatives.

Following results of McMullen concerning rational maps, we show that the limit set of matings between a certain class of representations of $C_2 \,\ast\, C_3$ and quadratic polynomials carries $\delta$-conformal measures, and that if the correspondence is

In E.N. Antoniou and S. Vologiannidis ( 2004), a new family of companion forms associated to a regular polynomial matrix has been presented generalizing similar results presented by M. Fiedler in M. Fiedler (2003) where the scalar case was considered. This family of companion forms preserves both the finite and infinite elementary divisors structure of the original polynomial matrix, thus all its members can be seen as linearizations of the corresponding polynomial matrix. In this note we examine its applications on polynomial matrices with symmetries which appear in a number of engineering fields

We study the behavior of discrete time AR-representations. A theorem is provided connecting the backward behavior of a system, due to its infinite elementary divisors, with the forward behavior of its dual system. We first use this result to construct a system satisfying a certain backward behavior. In addition to this, we propose a way to combine this result with previous ones to create an algorithm for computing a system satisfying a given forward and backward behavior.

Using the Rudin-Shapiro sequence, the existence of a regular but not strongly regular positive matrix that sums {exp(2irikt)} to 0 for all f E (0,1) is established.As a corollary it is shown that there exist matrices that sum all almost periodic sequences without possessing the Borel property and vice versa.1.A matrix A -(ank) of real or complex numbers is called strongly regular if it is regular and 00 (1) lim 2 K.fc -an.k+l\ = 0.Such matrices possess many interesting properties.Thus a matrix is translative if and only if it is strongly regular (cf.Petersen [5, p. 21]).Lorentz [3] has shown that a matrix sums every almost convergent sequence to its almost convergence limit if and only if it is strongly regular.If A = (an k) is strongly regular, then it necessarily satisfies the conditions CO (2) lim 2 an.k exp(27Tz/ci) = 0 and 00 (3) Hm 2 |fl.*lexp(27rifc) = 0 fl- 00 k=0for all / e (0, 1) as a simple Abel's transformation shows.The question naturally arises whether there exist regular matrices satisfying (2) and (3) that are not strongly regular.A positive answer to this question becomes much more significant if we observe that the Fourier-effective matrices

A 25-year-old and somewhat neglected algorithm of Crawford and Moon attempts to determine whether a given Hermitian matrix pair $(A,B)$ is definite by exploring the range of the function $f(x) = x^*(A+iB)x / | x^*(A+iB)x |$, which is a subset of the unit circle. We revisit the algorithm and show that with suitable modifications and careful attention to implementation details it provides a reliable and efficient means of testing definiteness. A clearer derivation of the basic algorithm is given that emphasizes an arc expansion viewpoint and makes no assumptions about the definiteness of the pair. Convergence of the algorithm is proved for all $(A,B$), definite or not. It is shown that proper handling of three details of the algorithm is crucial to the efficiency and reliability: how the midpoint of an arc is computed, whether shrinkage of an arc is permitted, and how directions of negative curvature are computed. For the latter, several variants of Cholesky factorization with complete pivoting are explored and the benefits of pivoting demonstrated. The overall cost of our improved algorithm is typically just a few Cholesky factorizations. Three applications of the algorithm are described: testing the hyperbolicity of a Hermitian quadratic matrix polynomial, constructing conjugate gradient methods for sparse linear systems in saddle point form, and computing the Crawford number of the pair $(A,B)$ via a quasiconvex univariate minimization problem.

ABSTRACTNew semi‐analytical shell elements are developed using the scaled boundary finite element method. A shell element is treated as a three‐dimensional continuum whose midsurface, the generalized “boundary” of the continuum, is characterized with a quadrilateral spectral element. It is along the thickness direction ξ that the midsurface is scaled to represent the three‐dimensional geometry and that the analytical displacement solution is sought. Neumann expansion is used to approximate the inverse of the Jacobian as a quadratic matrix polynomial of ξ while the assumed natural strain method is applied to alleviate the shear and membrane locking. The virtual work principle considering body forces is utilized to derive the scaled boundary finite element equation, which is directly solved via the differential quadrature method. Numerical examples show that the shell elements with five displacement sampling points along ξ can efficiently analyze thin to very thick general shells.

We consider even grade polynomial eigenvalue problems (PEP) to study their eigenvalue location and condition number bounds. We converted the even grade PEP into a Quadratic eigenvalue problem (QEP) of high dimension via quadratification techniques. We study an eigenvalue localization method that applies block Gersgorin sets to certain linearization classes of quadratic matrix polynomials originating from well-established vector spaces. We presented a few results on the bounding ratios of condition number of certain linearization classes with respect to that of original PEP and QEP induced due to quadratification. Relevant computational works are performed using a model numerical example to show the effectiveness of the results in terms of calculating eigenvalue and bounds of condition number ratios.