Geršgorin and His Circles

Varga, Richard S.

doi:10.1007/978-3-642-17798-9

Cited by 554 publications

(451 citation statements)

References 51 publications

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“…Hence, we have proven that for every i ∈ T (A), there exists a path in G(A) to some j ∈ T (A). ✷ The class of S-SDD matrices is the class of H-matrices introduced independently by Gao and Wang in 1992 ( [6]), and by Cvetković, Kostic and Varga in 2004 ( [7,2]). We use notation from [7,2].…”

Section: Characterization Of Diagonally Dominant H-matricesmentioning

confidence: 99%

“…✷ The class of S-SDD matrices is the class of H-matrices introduced independently by Gao and Wang in 1992 ( [6]), and by Cvetković, Kostic and Varga in 2004 ( [7,2]). We use notation from [7,2]. Definition 2 Given any matrix A = [a ij ] ∈ C n×n , n ≥ 2, and given any nonempty proper subset S of N , then A is an S-strictly diagonally dominant (S-SDD) if…”

Section: Characterization Of Diagonally Dominant H-matricesmentioning

confidence: 99%

“…The nonzero elements chains of A, mean that from every i ∈ T (A) there exists a path to some j ∈ T (A) in the directed graph G(A), associated to the matrix A (see [2]). …”

Section: Introduction and Notationmentioning

confidence: 99%

See 2 more Smart Citations

Characterization of diagonally dominant H-matrices

Morača

2017

Publ Inst Math (Belgr)

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We first show that sufficient conditions for a diagonally dominant matrix to be a nonsingular one (and also an H-matrix), obtained independently by Shivakumar andChew in 1974, andFarid in 1995, are equivalent. Then we simplify the characterization of diagonally dominant H-matrices obtained by Huang in 1995, and using it prove that the Shivakumar-Chew-Farid sufficient condition for a diagonally dominant matrix to be an H-matrix, is also necessary.

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Section: Characterization Of Diagonally Dominant H-matricesmentioning

confidence: 99%

Section: Characterization Of Diagonally Dominant H-matricesmentioning

confidence: 99%

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Characterization of diagonally dominant H-matrices

Morača

2017

Publ Inst Math (Belgr)

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“…Another type of eigenvalue bound for general matrices is a Gershgorin theorem [31], which in the simplest form states that for a diagonal matrix A = diag(λ 1 , . .…”

Section: Proof Fix An Index I and Letmentioning

confidence: 99%

Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices

Ipsen

Nadler²

2009

SIAM J. Matrix Anal. & Appl.

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Abstract. We present eigenvalue bounds for perturbations of Hermitian matrices, and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo arclength continuation methods for the solution of nonlinear systems.Key words. eigenvalues, Hermitian matrix, eigenvalue gap, perturbation bounds, non-Hermitian perturbations, principal components, numerical continuation.AMS subject classification. 65F15, 65H10, 65H17, 65H20, 15A18, 15A421. Introduction. We present perturbation bounds for eigenvalues of Hermitian matrices that were motivated by two applications: principal component analysis under a spiked covariance model [25], and pseudo arclength continuation methods for the solution of systems of nonlinear equations [7].Although these applications are very different, they share a common requirement: The change in the eigenvalues of interest should be determined not by the global norm of the full perturbation, which can be quite large, but rather by the norm of a projection of the perturbation on a particular eigenspace. In contrast, most existing eigenvalue bounds are expressed either in terms of the full perturbation or else in terms of a residual, and therefore do not provide sufficient information for our applications.The paper is organized as follows. We start with the most specific class of perturbations, Hermitian rank-one updates, and then generalize the perturbations first to Hermitian and then to non-Hermitian matrices. In §2 we present bounds for Hermitian rank one updates, and explain why such bounds can be useful in pseudo-arclength continuation methods. In §3 we consider Hermitian perturbations whose norm is smaller than the spectral gap of a specific eigenvalue, and describe their use in principal component analysis. In §4 we extend the bounds to non-Hermitian perturbations.

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“…Gershigorin circle theorem and Kharitonov theorem are well known methods to decide the stability of LTI systems with such uncertainties. Gershigorin approach provides the approximated bounds of system poles in the form of circles whose centers are the diagonal terms of a given system matrix and radii are the absolute summation of off-diagonal terms of the corresponding row [15]. Kharitanov approach provides several characteristic equations whose coefficients are the minimum and the maximum values of parametric variations and the stability is guaranteed only if all characteristic equations are stable [5,13].…”

Section: Introductionmentioning

confidence: 99%

Stability of LTI Systems with Unstructured Uncertainty Using Quadratic Disc Criterion

Yeom

Park²,

Joo

2012

Journal of Electrical Engineering and Technology

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-This paper deals with robust stability of linear time-invariant (LTI) systems with unstructured uncertainties. A new relation between uncertainties and system poles perturbed by the uncertainties is derived from a graphical analysis. A stability criterion for LTI systems with uncertainties is proposed based on this result. The migration range of the poles in the proposed criterion is represented as the bound of uncertainties, the condition number of a system matrix, and the disc containing the poles of a given nominal system. Unlike the existing methods depending on the solutions of algebraic matrix equations, the proposed criterion provides a simpler way which does not involves algebraic matrix equations, and a more flexible root clustering approach by means of adjusting the center and the radius of the disc as well as the condition number.

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Geršgorin and His Circles

Cited by 554 publications

References 51 publications

Characterization of diagonally dominant H-matrices

Characterization of diagonally dominant H-matrices

Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices

Stability of LTI Systems with Unstructured Uncertainty Using Quadratic Disc Criterion

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