Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds

@article{Eisenstat1998ThreeAP,
  title={Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds},
  author={Stanley C. Eisenstat and Ilse C. F. Ipsen},
  journal={SIAM J. Matrix Analysis Applications},
  year={1998},
  volume={20},
  pages={149-158}
}
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 absolute sense. We also show that eigenvalues of normal matrices are no more sensitive to perturbations… CONTINUE READING
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