Classroom Note: Some Eigenvalue Properties of Persymmetric Matrices

Reid, Russell M.

doi:10.1137/s0036144595294801

Cited by 38 publications

(19 citation statements)

References 1 publication

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“…Note that A is also a centrosymmetric matrix since A = F 2n AF 2n . A practically relevant example of such a matrix is the Gramian of a set of frequency exponentials {e ±ıλ k t }, which plays a role in the control of mechanical and electric vibrations [105]. Employing the orthogonal matrix U = 1…”

Section: Persymmetric Matricesmentioning

confidence: 99%

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Structured Eigenvalue Problems

Faßbender

Kreßner

2006

GAMM-Mitteilungen

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Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew‐symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Section: Persymmetric Matricesmentioning

confidence: 99%

“…Hence, the eigenvalues of A are the positive and negative square roots of the eigenvalues of the matrix product (A 11 F n + A 12 )(F n A 11 − A T 12 ), see also [105]. Structure-preserving Jacobi algorithms for symmetric persymmetric and skew-symmetric persymmetric matrices have been recently developed in [90].…”

Section: Persymmetric Matricesmentioning

confidence: 99%

Structured Eigenvalue Problems

Faßbender

Kreßner

2006

GAMM-Mitteilungen

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“…Following Reid [23] we define the "flip" operation ( ) F , whose effect is to transpose a matrix across its anti-diagonal:…”

Section: Flip Operatormentioning

confidence: 99%

“…In this paper we focus on four types of doubly structured real matricesthose that have symmetry or skew-symmetry about the anti-diagonal as well as the main diagonal. Instances where such matrices arise include the control of mechanical and electrical vibrations, where the eigenvalues and eigenvectors of Gram matrices that are symmetric about both diagonals play a fundamental role [23].…”

Section: Introductionmentioning

confidence: 99%

Structure preserving algorithms for perplectic eigenproblems

Mackey¹,

Mackey²,

Dunlavy³

2005

ELA

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Structured real canonical forms for matrices in R n×n that are symmetric or skew-symmetric about the anti-diagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solution of 4 × 4 subproblems constructed via quaternions, the algorithms calculate structured orthogonal bases for the invariant subspaces of the associated matrix. In addition to preserving structure, these methods are inherently parallelizable, numerically stable, and show asymptotic quadratic convergence.

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“…It will be convenient to use the "flip" operation [42], which transposes a matrix across its antidiagonal: A F def = = RA T R. 1. Double Givens: Let G denote a real 2 × 2 rotation.…”

Section: Givens-like Actionmentioning

confidence: 99%

Structured tools for structured matrices

Mackey¹,

Mackey²,

Tisseur³

2003

ELA

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Abstract. An extensive and unified collection of structure-preserving transformations is presented and organized for easy reference. The structures involved arise in the context of a nondegenerate bilinear or sesquilinear form on R n or C n . A variety of transformations belonging to the automorphism groups of these forms, that imitate the action of Givens rotations, Householder reflectors, and Gauss transformations are constructed. Transformations for performing structured scaling actions are also described. The matrix groups considered in this paper are the complex orthogonal, real, complex and conjugate symplectic, real perplectic, real and complex pseudo-orthogonal, and pseudo-unitary groups. In addition to deriving new transformations, this paper collects and unifies existing structure-preserving tools.

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Classroom Note: Some Eigenvalue Properties of Persymmetric Matrices

Cited by 38 publications

References 1 publication

Structured Eigenvalue Problems

Structured Eigenvalue Problems

Structure preserving algorithms for perplectic eigenproblems

Structured tools for structured matrices

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