NDLI: Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations

Please wait, while we are loading the content...

Structured matrix computations in non-Euclidean geometries: algorithms and applications : Editorial of the special issue dedicated to a workshop held at CIRM, Luminy (France), October 8–12, 2012

Fast computation of eigenvalues of companion, comrade, and related matrices

Minimization principles and computation for the generalized linear response eigenvalue problem

Geometric means of structured matrices

Implicit QR for rank-structured matrix pencils

Comparison of software for computing the action of the matrix exponential

The structure of iterative methods for symmetric linear discrete ill-posed problems

Refining estimates of invariant and deflating subspaces for large and sparse matrices and pencils

A projector-splitting integrator for dynamical low-rank approximation

A bisection method for measuring the distance of a quadratic matrix polynomial to the quadratic matrix polynomials that are singular on the unit circle

An algorithm for solving the indefinite least squares problem with equality constraints

Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations

Symmetric spaces and Lie triple systems in numerical analysis of differential equations

Equivalence between modified symplectic Gram-Schmidt and Householder SR algorithms

BIT Numerical Mathematics : Volume 53

BIT Numerical Mathematics : Volume 52

BIT Numerical Mathematics : Volume 51

BIT Numerical Mathematics : Volume 50

BIT Numerical Mathematics : Volume 49

BIT Numerical Mathematics : Volume 48

BIT Numerical Mathematics : Volume 47

BIT Numerical Mathematics : Volume 46

BIT Numerical Mathematics : Volume 45

BIT Numerical Mathematics : Volume 44

BIT Numerical Mathematics : Volume 43

BIT Numerical Mathematics : Volume 42

BIT Numerical Mathematics : Volume 41

BIT Numerical Mathematics : Volume 40

BIT Numerical Mathematics : Volume 39

BIT Numerical Mathematics : Volume 38

BIT Numerical Mathematics : Volume 37

Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations

Content Provider	Springer Nature Link
Author	Mehrmann, Volker Ran, André C. M. Mehl, Christian Rodman, Leiba
Copyright Year	2013
Abstract	We study the perturbation theory of structured matrices under structured rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic with respect to an indefinite inner product. The rank one perturbations are not necessarily of arbitrary small size (in the sense of norm). In the case of sesquilinear forms, results on selfadjoint matrices can be applied to unitary matrices by using the Cayley transformation, but in the case of real or complex symmetric or skew-symmetric bilinear forms additional considerations are necessary. For complex symplectic matrices, it turns out that generically (with respect to the perturbations) the behavior of the Jordan form of the perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan form of complex symplectic matrices are accounted for. For instance, the number of Jordan blocks of fixed odd size corresponding to the eigenvalue 1 or −1 have to be even. For complex orthogonal matrices, it is shown that the behavior of the Jordan structures corresponding to the original eigenvalues that are not moved by perturbations follows again the pattern established earlier for unstructured matrices, taking into account the specifics of Jordan forms of complex orthogonal matrices. The proofs are based on general results developed in the paper concerning Jordan forms of structured matrices (which include in particular the classes of orthogonal and symplectic matrices) under structured rank one perturbations. These results are presented and proved in the framework of real as well as of complex matrices.
Ending Page	255
Page Count	37
Starting Page	219
File Format	PDF
ISSN	00063835
e-ISSN	15729125
Journal	BIT Numerical Mathematics
Issue Number	1
Volume Number	54
Language	English
Publisher	Springer Netherlands
Publisher Date	2013-10-15
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Perturbation theory Orthogonal matrix Unitary matrix Computational Mathematics and Numerical Analysis Perturbation analysis Numeric Computing Indefinite inner product Mathematics Sensitivity (robustness) Rank one perturbation Cayley transformation Quadratic and bilinear forms, inner products Canonical forms, reductions, classification Generic perturbation Canonical structure Perturbations Symplectic matrix
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Mathematics Software

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Eigenvalue perturbation theory of structured matrices under generic structured rank one perturbations : Symplectic , orthogonal , and unitary matrices ∗

On rank one perturbations of Hamiltonian system with periodic coefficients

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence

Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

An informal introduction to perturbations of matrices determined up to similarity or congruence

Canonical matrices of bilinear and sesquilinear forms

A regularization algorithm for matrices of bilinear and sesquilinear forms

Change of the *congruence canonical form of 2-by-2 matrices under perturbations

Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations

Similar Documents

Eigenvalue perturbation theory of structured matrices under generic structured rank one perturbations : Symplectic , orthogonal , and unitary matrices ∗

On rank one perturbations of Hamiltonian system with periodic coefficients

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence

Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

An informal introduction to perturbations of matrices determined up to similarity or congruence

Canonical matrices of bilinear and sesquilinear forms

A regularization algorithm for matrices of bilinear and sesquilinear forms

Change of the *congruence canonical form of 2-by-2 matrices under perturbations

Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations