Perturbation theory for the eigenvalues of factorised symmetric matrices (original) (raw)

Relative perturbation theory for quadratic Hermitian 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 lambda2Mx+lambdaCx+Kx=0\lambda^2 M x + \lambda C x + K x = 0lambda2Mx+lambdaCx+Kx=0, where MMM and KKK are nonsingular Hermitian matrices and CCC is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair A−lambdaBA-\lambda BAlambdaB. 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.

Perturbation of Partitioned Hermitian Generalized Eigenvalue Problem

This paper is concerned with the Hermitian positive definite generalized eigenvalue problem A − λB for partitioned matrices A = A 11 A 22 , B = B 11 B 22 , where both A and B are Hermitian and B is positive definite. We present bounds on how its eigenvalues vary when A and B are perturbed by Hermitian matrices. These bounds are generally of linear order with respect to the perturbations in the diagonal blocks and of quadratic order with respect to the perturbations in the off-diagonal blocks. The results for the case of no perturbations in the diagonal blocks can be used to bound the changes of eigenvalues of a Hermitian positive definite generalized eigenvalue problem after its off-diagonal blocks are dropped, a situation that occurs frequently in eigenvalue computations. The presented results extend those of Li and Li (Linear Algebra Appl., 395 (2005), pp.183– 190). It was noted in Stewart and Sun (Matrix Perturbation Theory, Academic Press, Boston, 1990, p.300.) that different co...

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.

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.

Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbations

Numerical Algorithms, 2008

In this paper we address the problem of efficiently computing all the eigenvalues of a large N × N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N .

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.

Weyl-type relative perturbation bounds for eigensystems of Hermitian matrices

Linear Algebra and its Applications, 2000

We present a Weyl-type relative bound for eigenvalues of Hermitian perturbations A + E of (not necessarily definite) Hermitian matrices A. This bound, given in function of the quantity η = A −1/2 EA −1/2 2 , that was already known in the definite case, is shown to be valid as well in the indefinite case. We also extend to the indefinite case relative eigenvector bounds which depend on the same quantity η. As a consequence, new relative perturbation bounds for singular values and vectors are also obtained. Using matrix differential calculus techniques we obtain for eigenvalues a sharper, first-order bound involving the logarithm matrix function, which is smaller than η not only for small E, as expected, but for any perturbation.