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Stochastic Perturbation Theory
, 1988
"... . In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variatio ..."
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Cited by 617 (31 self)
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. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variation in the perturbed quantity. Up to the higherorder terms that are ignored in the expansion, these statistics tend to be more realistic than perturbation bounds obtained in terms of norms. The technique is applied to a number of problems in matrix perturbation theory, including least squares and the eigenvalue problem. Key words. perturbation theory, random matrix, linear system, least squares, eigenvalue, eigenvector, invariant subspace, singular value AMS(MOS) subject classifications. 15A06, 15A12, 15A18, 15A52, 15A60 1. Introduction. Let A be a matrix and let F be a matrix valued function of A. Two principal problems of matrix perturbation theory are the following. Given a matrix E, pr...
Relative Perturbation Results For Eigenvalues And Eigenvectors Of Diagonalisable Matrices
, 1996
"... . Let and x be a perturbed eigenpair of a diagonalisable matrix A. The problem is to bound the error in and x. We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound implies that the condition number for x is the norm of an orthogonal ..."
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Cited by 7 (2 self)
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. Let and x be a perturbed eigenpair of a diagonalisable matrix A. The problem is to bound the error in and x. We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound implies that the condition number for x is the norm of an orthogonal projection of the reduced resolvent at . This condition number can be a lot less pessimistic than the traditional one, which is derived from a firstorder analysis. A further upper bound leads to an extension of Davis and Kahan's sin ` Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and x are an exact eigenpair of a perturbed matrix D1AD2 , where D1 and D2 are nonsingular, but D1AD2 is not necessarily diagonalisable. We derive a bound on the relative error in and a sin ` theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation of D 1 and D2 from similarity and the deviation of D2 from iden...
Absolute And Relative Perturbation Bounds For Invariant Subspaces Of Matrices
, 1998
"... . Absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices, in the twonorm and in the Frobenius norm. The absolute bounds are extensions of Davis and Kahan's sin ` theorem to general matrices and invariant subspaces of any dimension. Wh ..."
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Cited by 4 (1 self)
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. Absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices, in the twonorm and in the Frobenius norm. The absolute bounds are extensions of Davis and Kahan's sin ` theorem to general matrices and invariant subspaces of any dimension. When the perturbed subspace has dimension one, the relative bound is implied by the absolute bound. The relative bounds presented here are the most general relative bounds for invariant subspaces because they place no restrictions on the matrix or the perturbation. Key words. invariant subspace, condition number, separation between matrices, absolute error, relative error, eigenvalue separation, angle between subspaces AMS subject classification. 15A12, 15A18, 15A42, 15A69, 65F15, 65F35 1. Introduction. Absolute and relative perturbation bounds are derived for angles between invariant subspaces of a complex square matrix A and a perturbed matrix A+E, in the twonorm and in the Frobenius no...