Journal ArticleDOI
Perturbation bounds in connection with singular value decomposition
TLDR
The sin ϑ theorem for Hermitian linear operators in Davis and Kahan as discussed by the authors is applicable to computational solution of overdetermined systems of linear equations and especially cover the rank deficient case when the matrix is replaced by one of lower rank.Abstract:
LetA be anm ×n-matrix which is slightly perturbed. In this paper we will derive an estimate of how much the invariant subspaces ofA
H
A andAA
H
will then be affected. These bounds have the sin ϑ theorem for Hermitian linear operators in Davis and Kahan [1] as a special case. They are applicable to computational solution of overdetermined systems of linear equations and especially cover the rank deficient case when the matrix is replaced by one of lower rank.read more
Citations
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References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Journal ArticleDOI
The Rotation of Eigenvectors by a Perturbation. III
Chandler Davis,William Kahan +1 more
TL;DR: In this article, the difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other, and Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation or a computable residual.
Journal ArticleDOI
Perturbation bounds for means of eigenvalues and invariant subspaces
TL;DR: In this article, the authors derived bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix.
Journal ArticleDOI
Ill-conditioned systems of linear algebraic equations
TL;DR: In this article, a method called the method of false perturbations is proposed for the solution of ill-conditioned systems, which gives the solution x of equation (1) in the form of an orthogonal sum x(l) + xc2, where the term x (l) is extremely sensitive both to inherent (ineradicable according to the terminology of [l]) errors as well as to errors of rounding off, while the second term has no such sensitivity.