Absolute and relative Weyl theorems for generalized eigenvalue problems (original) (raw)
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.
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 BA−lambdaB. 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...
Optimal perturbation bounds for the Hermitian eigenvalue problem
Linear Algebra and its Applications, 2000
There is now a large literature on structured perturbation bounds for eigenvalue problems of the form H x = λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λ i , of the form * Corresponding author.
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.
A relative perturbation bound for positive definite matrices
Linear Algebra and its Applications, 1998
We give a sharp estimate for the eigenvectors of a positive definite Hermitian matrix under a floating-point perturbation. The proof is elementary. Recently there have been a number of papers on eigenvector perturbation bounds that involve a perturbation of the matrix which is small in some relative sense, including the typical rounding errors in matrix elements ( . Some of these have complicated proofs and all of them involve the notion of 'the relative gap' between the eigenvalues. i.e. a relative distance of the unperturbed eigenvalue to the rest of the spectrum. Several such relative gaps are in use. Anyway, in any such estimate it is only the nearest eigenvalue that matters, one does not care for distant eigenvalues and their influence. Our bounds control primarily the matrix of the angles between the perturbed and the unperturbed eigenvectors, standard bounds with relative gaps may be derived from them any time. In particular, in our bounds the distant eigenvalues naturally damp out the perturbation of the corresponding components of the eigenvector. Or bounds are asymptotically sharp i.e. for small perturbations they reach the first term of the perturbation theory. Our proof is simple (of all works cited above ([4]) is closest to our idea) -its only technical tool is taking the square root of a positive definite matrix. The simplicity of our proof may make it useful in a classroom.
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...
Perturbed eigenvalue problems: an overview
Studia Universitatis Babes-Bolyai Matematica, 2021
The study of perturbed eigenvalue problems has been a very active field of investigation throughout the years. In this survey we collect several results in the field.
Non-linear bounds for the generalized eigensystem of a matrix pencil with distinct eigenvalues
In this paper a non-local sensitivity analysis of the generalized eigen-system (eigenvectors and eigenvalues) of an n n matrix pencil A B with pairwise distinct eigenvalues is made. By rewriting the perturbed problem as an operator equation and using the technique of Lyapunov majorants non-local non-linear bounds are established. An example demonstrates the e ectiveness of the estimates proposed.
Perturbation theory for the eigenvalues of factorised symmetric matrices
Linear Algebra and its Applications, 2000
We obtain eigenvalue perturbation results for a factorised Hermitian matrix H = GJ G * where J 2 = I and G has full row rank and is perturbed into G + δG, where δG is small with respect to G. This complements the earlier results on the easier case of G with full column rank. Applied to square factors G our results help to identify the so-called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices.
The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach
2005
This work focuses on non-square matrix pencils A − λB where A, B ∈ M m×n and m > n. Traditional methods for solving such non-square generalized eigenvalue problems (A − λB)v = 0 are expected to lead to no solutions in most cases. In this paper we propose a different treatment, we search for the minimal perturbation on the pair {A, B} such that these solutions are indeed possible. Two cases are considered and analyzed: (i) The case where n = 1 (vector pencils); and (ii) more generally n > 1 case with the existence of one eigenpair. For both, this paper proposes insight into the characteristics of the described problems along with practical numerical algorithms towards their solution. We present also a new factorization for such non-square pencils, and some relations with the notion of pseudospectra.
Linear Algebra and its Applications, 2006
For a symmetric positive semi-definite diagonally dominant matrix, if its off-diagonal entries and its diagonally dominant parts for all rows (which are defined for a row as the diagonal entry subtracted by the sum of absolute values of off-diagonal entries in that row) are known to a certain relative accuracy, we show that its eigenvalues are known to the same relative accuracy. Specifically, we prove that if such a matrix is perturbed in a way that each off-diagonal entry and each diagonally dominant part have relative errors bounded by some , then all its eigenvalues have relative errors bounded by . The result is extended to the generalized eigenvalue problem.
Floating-point perturbations of Hermitian matrices
Linear Algebra and its Applications, 1993
We consider the perturbation properties of the eigensolution of Hermitian matrices. For the matrix entries and the eigenvalues we use the realistic "floatingpoint" error measure l&/al. Recently, Demmel and Veselid considered the same problem for a positive definite matrix H, showing that the floating-point perturbation theory holds with constants depending on the condition number of the matrix A = DHD, where Aij = 1 and D is a diagonal scaling. We study the general Hermitian case along the same lines, thus obtaining new classes of well-behaved matrices and matrix pairs. Our theory is applicable to the already known class of scaled diagonally dominant matrices as well as to matrices given by factors-like those in symmetric indefinite decompositions. We also obtain norm estimates for the perturbations of the eigenprojections, and show that some of our techniques extend to non-Hermitian matrices. However, unlike in the positive definite case, we are still unable to describe simply the set of all well-behaved Hermitian matrices. 1.
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.
On relative residual bounds for the eigenvalues of a Hermitian matrix
1996
Let H be a Hermitian matrix, X an orthonormal matrix, and M = X*HX. Then the eigenvalues of M approximate some eigenvalues of H with an absolute error bounded by I]ttX-XMI]2. The main interest in this work is the relative distance between the eigenvalues of M and some part of the spectrum of H. It is shown that distance depends on the angle between the ranges of X and HX.