Relative Perturbation Theory for Matrix Spectral Decompositions

Dissertation Zagreb, 2000. to Branka Acknowledgement I would like to express my thanks to all who have helped and supported me during the writing of this thesis. First of all I would like to thank my mentor Prof. Dr. Ivan Slapničar for introducing me to the exciting field of relative perturbation theory, for devoting to me a lot of his time, and for sharing with me so many of his ideas. I also wish to thank Prof. Dr.

[1]  Ilse C. F. Ipsen,et al.  Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices , 1998 .

[2]  Ji-guang Sun Eigenvalues of Rayleigh quotient matrices , 1991 .

[3]  Froilán M. Dopico,et al.  Weyl-type relative perturbation bounds for eigensystems of Hermitian matrices , 2000 .

[4]  J. Barlow,et al.  Computing accurate eigensystems of scaled diagonally dominant matrices: LAPACK working note No. 7 , 1988 .

[5]  Ilse C. F. Ipsen,et al.  Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds , 1998, SIAM J. Matrix Anal. Appl..

[6]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[7]  W. Gragg,et al.  On computing accurate singular values and eigenvalues of acyclic matrices , 1992 .

[8]  Ren-Cang Li,et al.  Relative Perturbation Theory: II. Eigenspace and Singular Subspace Variations , 1996, SIAM J. Matrix Anal. Appl..

[9]  Chandler Davis The rotation of eigenvectors by a perturbation , 1963 .

[10]  K. Veselié A Jacobi eigenreduction algorithm for definite matrix pairs , 1993 .

[11]  Shmuel Friedland,et al.  Singular values, doubly stochastic matrices, and applications , 1995 .

[12]  Ren-Cang Li Relative Perturbation Theory: I. Eigenvalue and Singular Value Variations , 1998, SIAM J. Matrix Anal. Appl..

[13]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[14]  Ilse C. F. Ipsen An overview of relative sin T theorems for invariant subspaces of complex matrices , 2000 .

[15]  Z. Drmač,et al.  Relative Residual Bounds For The Eigenvalues of a Hermitian Semidefinite Matrix , 1997 .

[16]  Conditioning in the Application of-orthogonal Transformations , 1996 .

[17]  Jesse L. Barlow,et al.  Optimal perturbation bounds for the Hermitian eigenvalue problem , 2000 .

[18]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[19]  P. Wedin On angles between subspaces of a finite dimensional inner product space , 1983 .

[20]  K. Veseli,et al.  Perturbation Theory for the Eigenvalues of Factorised Symmetric Matrices , 1999 .

[21]  I. Dhillon Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem , 1998 .

[22]  K. Veselic On a new class of elementary matrices , 1979 .

[23]  Roy Mathias,et al.  Quadratic Residual Bounds for the Hermitian Eigenvalue Problem , 1998 .

[24]  Leiba Rodman,et al.  Errata for: Polar decomposition in finite dimensional indefinite scalar product spaces: Special cases and applications , 1997 .

[25]  Ivan Slapničar,et al.  Perturbations of the eigenprojections of a factorized Hermitian matrix , 1995 .

[26]  Z. Drmač Accurate Computation of the Product-Induced Singular Value Decomposition with Applications , 1998 .

[27]  G. Stewart Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[28]  James Demmel,et al.  Accurate Singular Values of Bidiagonal Matrices , 1990, SIAM J. Sci. Comput..

[29]  Roy Mathias,et al.  A relative perturbation bound for positive definite matrices , 1998 .

[30]  Ilse C. F. Ipsen Absolute and relative perturbation bounds for invariant subspaces of matrices , 2000 .

[31]  Ivan Slapničar,et al.  Floating-point perturbations of Hermitian matrices , 1993 .

[32]  C. Loan Generalizing the Singular Value Decomposition , 1976 .

[33]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[34]  Ji-guang Sun The perturbation bounds for eigenspaces of a definite matrix-pair , 1983 .

[35]  J. Demmel,et al.  Computing the Singular Value Decomposition with High Relative Accuracy , 1997 .

[36]  Ji-guang Sun,et al.  Perturbation analysis for the generalized eigenvalue and the generalized singular value problem , 1983 .

[37]  H. Zha A note on the existence of the hyperbolic singular value decomposition , 1996 .

[38]  James Demmel,et al.  Jacobi's Method is More Accurate than QR , 1989, SIAM J. Matrix Anal. Appl..

[39]  Tosio Kato Perturbation theory for linear operators , 1966 .

[40]  A. Bojanczyk,et al.  The hyperbolic singular value decomposition and applications , 1989, Proceedings of the 32nd Midwest Symposium on Circuits and Systems,.

[41]  Zlatko Drmač On relative residual bounds for the eigenvalues of a Hermitian matrix , 1996 .

[42]  A. Sluis Condition numbers and equilibration of matrices , 1969 .

[43]  Ivan Slapničar,et al.  Componentwise Analysis of Direct Factorization of Real Symmetric and Hermitian Matrices , 1998 .

[44]  R. Mathias Spectral Perturbation Bounds for Positive Definite Matrices , 1997 .

[45]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[46]  B. Parlett The Symmetric Eigenvalue Problem , 1981 .

[47]  M. Saunders,et al.  Towards a Generalized Singular Value Decomposition , 1981 .

[48]  Ivan Slapničar,et al.  Accurate Symmetric Eigenreduction by a Jacobi Method , 1993 .

[49]  Allan O. Steinhardt,et al.  The hyberbolic singular value decomposition and applications , 1990, Fifth ASSP Workshop on Spectrum Estimation and Modeling.

[50]  Ilse C. F. Ipsen,et al.  Relative perturbation techniques for singular value problems , 1995 .

[51]  Ilse C. F. Ipsen Relative perturbation results for matrix eigenvalues and singular values , 1998, Acta Numerica.

[52]  Ivan Slapničar,et al.  Relative perturbation bound for invariant subspaces of graded indefinite Hermitian matrices , 1999 .