On error bounds for eigenvalues of a matrix pencil (original) (raw)
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.
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...
Bounds for characteristic values of entire matrix pencils
Linear Algebra and its Applications, 2004
Entire matrix-valued functions of a complex argument (entire matrix pencils) are considered. Upper bounds for sums of characteristic values and a lower bound for the smallest characteristic value are derived in the terms of the coefficients of the Taylor series. These results are new even for polynomial pencils.
SIAM Journal on Matrix Analysis and Applications, 2014
We derive a formula for the backward error of a complex number λ when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, *-even, and *-odd. Numerical experiments suggest that in many cases there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.
On variations of characteristic values of entire matrix pencils
Journal of Approximation Theory, 2005
Entire matrix-valued functions of a complex argument (entire matrix pencils) are considered. Bounds for spectral variations of pencils are derived. In particular, approximations of entire pencils by polynomial pencils are investigated. Our results are new even for polynomial pencils.
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.
Numerical range of linear pencils
Linear Algebra and its Applications, 2000
Consider a linear pencil Aλ + B, where A and B are n × n complex matrices. The numerical range of Aλ + B is defined as
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.