We prove that the 2-norm distance from an nn matrix A to the matrices that have a multiple eigenvalue is equal to rsep (A) = max 0 2n?1
We discuss two spectral dichotomy techniques: one for computing an invariant subspace of a nonsymmetric matrix associated with the eigenvalues inside and outside a given parabola. Another for computing a right deeating subspace of a regular matrix pencil associated with the eigenvalues inside and outside a given ellipse. The techniques use matrices of order… (More)
We provide several optimal backward perturbation bounds for the linear least squares problem with a matrix of deecient rank whose solution is deened by means of the truncated singular value decomposition.
Peters and Wilkinson 4] state that \it is well known that Gauss-Jordan is stable" for a diagonally dominant matrix, but a proof does not seem to have been published 3]. The present note lls this gap. Gauss-Jordan elimination is backward stable for matrices diagonally dominant by rows and not backward stable for matrices diagonally dominant by columns. In… (More)
SUMMARY A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating sub-spaces corresponding to the… (More)
We prove that the 2-distance from an n n matrix A to the matrices that have a multiple eigenvalue is equal to max 0 2n?1
K.V. Fernando developed an efficient approach for computation of an eigenvector of a tridiagonal matrix corresponding to an approximate eigenvalue. We supplement Fernando's method with deflation procedures by Givens rotations. These deflations can be used in the Lanczos process and instead of the inverse iteration.