Relative perturbation theory: (I) eigenvalue variations

In this paper, we consider how eigenvalues of a matrix $A$ change when it is perturbed to $\wtd A=D_1^*AD_2$ and how singular values of a (nonsquare) matrix $B$ change when it is perturbed to $\wtd B=D_1^*BD_2$, where $D_1$ and $D_2$ are assumed to be close to unitary matrices of suitable dimensions. We have been able to generalize many well-known perturbation theorems, including Hoffman-Wielandt theorem and Weyl-Lidskii theorem. As applications, we obtained bounds for perturbations of graded matrices in both singular value problems and nonnegative definite Hermitian eigenvalue problems.

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

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

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

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  C. T. Fike,et al.  Norms and exclusion theorems , 1960 .

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

[7]  Singular Values and Doubly Stochastic Matrices , 1993 .

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

[9]  J. Demmel,et al.  The bidiagonal singular value decomposition and Hamiltonian mechanics: LAPACK working note No. 11 , 1989 .

[10]  G. Stewart,et al.  A block QR algorithm and the singular value decomposition , 1993 .

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

[12]  L. Mirsky.  SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS , 1960 .

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

[14]  William Kahan,et al.  Spectra of nearly Hermitian matrices , 1975 .

[15]  B. Parlett,et al.  Accurate singular values and differential qd algorithms , 1994 .

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

[17]  Rajendra Bhatia,et al.  Some inequalities for commutators and an application to spectral variation , 1990 .

[18]  Karl Löwner.  Über monotone Matrixfunktionen , 1934 .

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

[20]  G. Forsythe,et al.  The proper values of the sum and product of symmetric matrices , 1953 .

[21]  H. Wielandt.  An extremum property of sums of eigenvalues , 1955 .

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

[23]  A. Hoffman,et al.  The variation of the spectrum of a normal matrix , 1953 .

[24]  W. Kahan.  Accurate eigenvalues of a symmetric tri-diagonal matrix , 1966 .

[25]  H. Weyl.  Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) , 1912 .

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

[27]  Ren-Cang Li,et al.  Bounds on perturbations of generalized singular values and of associated subspaces , 1993 .