Non-linear bounds for the generalized eigensystem of a matrix pencil with distinct eigenvalues (original) (raw)

Sensitivity and Accuracy of Eigenvalues Relative to Their Perturbation

The main objective of this paper is to study the sensitivity of eigenvalues in their computational domain under perturbations, and to provide a solid intuition with some numerical example as well as to represent them in graphically. The sensitivity of eigenvalues, estimated by the condition number of the matrix of eigenvectors has been discussed with some numerical example. Here, we have also demonstrated, other approaches imposing some structures on the complex eigenvalues, how this structure affects the perturbed eigenvalues as well as what kind of paths do they follow in the complex plane.

On sensitivity of eigenvalues and eigendecompositions of matrices

Linear Algebra and its Applications, 2005

The main purpose of this paper is to provide solutions of two problems on sensitivity of eigenvalues and eigendecompositions of matrices. The first problem is due to Wilkinson and it concerns finding the distance from an n × n matrix with n distinct eigenvalues to the set of matrices having multiple eigenvalues. We also describe how to construct a nearest matrix having a multiple eigenvalue. The second problem concerns providing a characterization of the stability of eigendecompositions of matrices and is due to Demmel.

The sensitivity of eigenvalues under elementary matrix perturbations

Linear Algebra and its Applications, 1987

Let M be an n X n real matrix, and let E,, be the elementary matrix with 1 in the (x, y) position and zero elsewhere. For L E C we call the matrix M + .zE,, an elementary matrix perturbation of M. Let X be any eigenvalue of M. Then there exists an (x, y) pair, 1~ x, y < n, and an analytic function h,,(z) defined in a neighborhood N of the origin such that: (a) h,,(O) = X. (b) h,,(z) is an eigenvalue of the elementary matrix perturbation M + .z~(~)E_, for any z E N, where k(X) is the dimension of the largest block containing X in the Jordan canonical form of M. (c) For any z E N, .z # 0, M + zE,, has k(X) distinct eigenvalues, all different from X. If X(z) is any one of these, then 1X -h(z)1 = O(l~li/~(~)). (d) For any z E N, .z # 0, M + zE,, has eigenvalue X with multiplicity s(A) -k(X), where s(X) is the (algebraic) multiplicity of X in M. (e) For all real positive or negative t E N, but

Relative Perturbation Theory for Quadratic Eigenvalue Problems

arXiv: Numerical Analysis, 2016

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form λ 2 Mx + λCx + Kx = 0, where M and K are nonsingular Hermitian matrices and C is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair A − λB. The new bounds can be applied to many interesting quadratic eigenvalue problems appearing in applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.

Generalized eigenvalue problems with specified eigenvalues

IMA Journal of Numerical Analysis, 2014

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in ] regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of BFGS and Lipschitz-based global optimization algorithms.

A general framework for the perturbation theory of matrix equations

2021

A general framework is presented for the local and non-local perturbation analysis of general real and complex matrix equations in the form F(P,X)=0F(P,X) = 0F(P,X)=0, where FFF is a continuous, matrix valued function, PPP is a collection of matrix parameters and XXX is the unknown matrix. The local perturbation analysis produces condition numbers and improved first order homogeneous perturbation bounds for the norm ∣deX∣\|\de X\|deX or the absolute value ∣deX∣|\de X|deX of deX\de XdeX. The non-local perturbation analysis is based on the method of Lyapunov majorants and fixed point principles. % for the operator pi(p,cdot)\pi(p,\cdot)pi(p,cdot). It gives rigorous non-local perturbation bounds as well as conditions for solvability of the perturbed equation. The general framework can be applied to various matrix perturbation problems in science and engineering. We illustrate the procedure with several simple examples. Furhermore, as a model problem for the new framework we derive a new perturbation theory for continuous-time algebr...

Rank one perturbation with a generalized eigenvector

arXiv: Spectral Theory, 2020

The relationship between the Jordan structures of two matrices sufficiently close has been largely studied in the literature, among which a square matrix A and its rank one updated matrix of the form A+xb * are of special interest. The eigenvalues of A + xb * , where x is an eigenvector of A and b is an arbitrary vector, were first expressed in terms of eigenvalues of A by Brauer in 1952. Jordan structures of A and A+xb * have been studied, and similar results were obtained when a generalized eigenvector of A was used instead of an eigenvector. However, in the latter case, restrictions on b were put so that the spectrum of the updated matrix is the same as that of A. There does not seem to be results on the eigenvalues and generalized eigenvectors of A + xb * when x is a generalized eigenvector and b is an arbitrary vector. In this paper we show that the generalized eigenvectors of the updated matrix can be written in terms of those of A when a generalized eigenvector of A and an arbitrary vector b are involved in the perturbation.

Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey

Axioms, 2021

The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which refines the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, ta...

Perturbation analysis on matrix pencils for two specified eigenpairs of a semisimple eigenvalue with multiplicity two

ETNA - Electronic Transactions on Numerical Analysis

In this paper, we derive backward error formulas of two approximate eigenpairs of a semisimple eigenvalue with multiplicity two for structured and unstructured matrix pencils. We also construct the minimal structured perturbations with respect to the Frobenius norm such that these approximate eigenpairs become exact eigenpairs of an appropriately perturbed matrix pencil. The structures we consider include T-symmetric/T-skewsymmetric, Hermitian/skew-Hermitian, T-even/T-odd, and H-even/H-odd matrix pencils. Further, we establish various relationships between the backward error of a single approximate eigenpair and the backward error of two approximate eigenpairs of a semisimple eigenvalue with multiplicity two.