Review of perturbation bounds for matrix eigenvalues by R. Bhatia

Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds

We show that three well-known perturbation bounds for matrix eigenvalues imply relative bounds: the Bauer-Fike and Hooman-Wielandt theorems for diagonalisable matrices, and Weyl's theorem for Hermitian matrices. As a consequence, relative perturbation bounds are not necessarily stronger than absolute bounds; and the conditioning of an eigenvalue in the relative sense is the same as in the absol...

Bounds for eigenvalues of matrix polynomials

Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block ...

Perturbation bounds for eigenvalues of diagonalizable matrices and singular values

Perturbation Bounds for Hyperbolic Matrix Factorizations

Several matrix factorizations depend on orthogonal factors, matrices that preserve the Euclidean scalar product. Some of these factorizations can be extended and generalized to (J, J̃)-orthogonal factors, that is, matrices that satisfy H JH = J̃ , where J and J̃ are diagonal with diagonal elements ±1. The purpose of this work is to analyze the perturbation of matrix factorizations that have a (J, ...

Accurate Error Bounds for the Eigenvalues of the Kernel Matrix

The eigenvalues of the kernel matrix play an important role in a number of kernel methods, in particular, in kernel principal component analysis. It is well known that the eigenvalues of the kernel matrix converge as the number of samples tends to infinity. We derive probabilistic finite sample size bounds on the approximation error of individual eigenvalues which have the important property th...