Localization of eigenvalues of polynomial matrices (original) (raw)

Spectrum localization of regular matrix polynomials and functions

The Electronic Journal of Linear Algebra, 2010

This paper is devoted to the spectrum localization problem for regular matrix polynomials and functions. Sufficient conditions are proposed for spectrum placement in a wide class of regions bounded by analytical curves. These conditions generalize the known linear matrix inequalities (LMI) approaches to stability analysis and pole placement of polynomial matrices. In addition, a method of robust spectrum placement is developed in the form of the LMI systems for a parametric set of matrix polynomials.

On the location of eigenvalues of matrix polynomials

Operators and Matrices, 2019

A number λ ∈ C is called an eigenvalue of the matrix polynomial P (z) if there exists a nonzero vector x ∈ C n such that P (λ)x = 0. Note that each finite eigenvalue of P (z) is a zero of the characteristic polynomial det(P (z)). In this paper we establish some (upper and lower) bounds for eigenvalues of matrix polynomials based on the norm of their coefficient matrices and compare these bounds to those given by N.J. Higham and F. Tisseur [8], J. Maroulas and P. Psarrakos [12].

On condition numbers of polynomial eigenvalue problems

Applied Mathematics and Computation, 2010

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

An eigenvalue inequality and spectrum localization for complex matrices

Using the notions of the numerical range, Schur complement and unitary equivalence, an eigenvalue inequality is obtained for a general complex matrix, giving rise to a region in the complex plane that contains its spectrum. This region is determined by a curve, generalizing and improving classical eigenvalue bounds obtained by the Hermitian and skew-Hermitian parts, as well as the numerical range of a matrix.

Localization of the spectrum and representation of solutions of linear dynamical systems

Ukrainian Mathematical Journal, 1998

We develop a general method for the localization of eigenvalues of matrix polynomials and functions based on the solution of matrix equations. For a broad class of equations, we formulate theorems that generalize the known properties of the Lyapunov equation. A new method for the representation of solutions of linear differential and difference systems is proposed.

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.

Localization of generalized eigenvalues by Cartesian ovals

Numerical Linear Algebra with Applications, 2011

In this paper, we consider the localization of generalized eigenvalues, and we discuss ways in which the Gersgorin set for generalized eigenvalues can be approximated. Earlier, Stewart proposed an approximation using a chordal metric. We will obtain here an improved approximation, and using the concept of generalized diagonal dominance, we prove that the new approximation has some of the basic properties of the original Geršgorin set, which makes it a handy tool for generalized eigenvalue localization. In addition, an isolation property is proved for both the generalized Geršgorin set and its approximation.

The boundary of the numerical range of matrix polynomials

Linear Algebra and its Applications, 1997

Some algebraic properties of the sharp points of the numerical range of matrix polynomials are the main subject of this paper. We also consider isolated points of the numerical range and the location of the numerical range in a circular annulus. 0 1997 Elsevier Science Inc.

Localization of spectrum and stability of certain classes of dynamical systems

Ukrainian Mathematical Journal, 1996

UDC 517.512 We develop a method for the localization of spectra of multiparameter matrix pencils and matrix functions, which reduces the problem to the solution of linear matrix equations and inequalities. We formulate algebraic conditions for the stability of linear systems of differential, difference, and difference-differential equations.

On the connectedness of numerical range of matrix polynomials

Linear Algebra and its Applications, 1998

An investigation on nonconnectedness of numerical range for manic matrix polynomials L(1) is undertaking here. The distribution of eigenvalues of L(1) to the components of numerical range and some other algebraic properties are also presented.