Bounds on the error of an approximate invariant subspacefor non-self-adjoint matrices (original) (raw)

Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices – Application of perturbation theory for simple invariant subspaces

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.

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.

Relative Perturbation Bound for Invariant Subspaces of Hermitian Matrix

Glasnik Matematicki, 2000

Abstract. We give a bound for the perturbations of invariant sub-spaces of a non-singular Hermitian matrix H under relative additive per-turbations of H. Such perturbations include the case when the elements ofH are known up to some relative tolerance. Our bound is, in appropriatecases, sharper than the classical bounds, and it generalizes some of therecent relative perturbation results. 1. Introduction and preliminariesWe consider the Hermitian eigenvalue problemH= Q Q =X ni=1  i q i q ;where H is a non-singular Hermitian matrix of order n, = diag( i ) is adiagonal matrix whose diagonal elements are the eigenvalues of H, and Q=q 1 q 2 q n isanunitarymatrixwhosei-th columnisthe eigenvectorwhich corresponds to  i . We denote the set of all eigenvalues of Hby ˙(H) =f 1 ; ; n g. We also assume that the eigenvalues are ordered,  1   2    n .Subspace X is an invariant subspace of a general matrix Hif HX  X.We consider invariant subspaces which correspond to the set of kneighb...

Best Approximations by Increasing Invariant Subspaces of Self-Adjoint Operators

2020

The paper describes approximations properties of monotonically increasing sequences of invariant subspaces of a self-adjoint operator, as well as their symmetric generalizations in a complex Hilbert space, generated by its positive powers. It is established that the operator keeps its spectrum over the dense union of these subspaces, equipped with quasi-norms, and that it is contractive. The main result is an inequality that provides an accurate estimate of errors for the best approximations in Hilbert spaces by these invariant subspaces.

Relative Perturbation Bound for Invariant Subspaces of Indefinite Hermitian Matrix

1998

We give bound for the perturbations of invariant subspaces of nonsingular indefinite Hermitian matrix H under relative additive perturbations of H. Such perturbations include the case when the elements of H are known up to some relative tolerance. Our bound is, in appropriate cases, sharper than the classical bounds, and it generalises some of the recent relative perturbation results. Mathematics subject classification (1991): 15A42, 65F35, 15A60. Key words and phrases: Hermitian matrix, invariant subspace, spectral projection, perturbation bound. 1 Introduction and preliminaries We consider the Hermitian eigenvalue problem H = QQ = n X i=1 i q i q i ; where H is a non-singular Hermitian matrix of order n, = diag( i ) is a diagonal matrix whose diagonal elements are the eigenvalues of H, and Q = [ q 1 q 2 \Delta \Delta \Delta q n ] is unitary matrix whose i-th column is the eigenvector which corresponds to i . Throughout the paper we assume that the eigenvalues are order...

Relative perturbation bound for invariant subspaces of Hermitian matrices

1999

Abstract. We give a bound for the perturbations of invariant sub-spaces of a non-singular Hermitian matrix H under relative additive per-turbations of H. Such perturbations include the case when the elements of H are known up to some relative tolerance. Our bound is, in appropriate cases, sharper than the classical bounds, and it generalizes some of the recent relative perturbation results.

On matrix approximation

Proceedings of the American Mathematical Society, 1975

In this paper we give an algebraic characterization of the best approximants to a given matrix A from a real line spanned by a matrix B. The distance | A-aßfl is taken to be the spectral norm of A-aB.

On a Perturbation Bound for Invariant Subspaces of Matrices

SIAM Journal on Matrix Analysis and Applications, 2014

Given a nonsymmetric matrix A, we investigate the effect of perturbations on an invariant subspace of A. The result derived in this paper differs from Stewart's classical result and sometimes yields tighter bounds. Moreover, we provide norm estimates for the remainder terms in well-known perturbation expansions for invariant subspaces, eigenvectors, and eigenvalues.

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...