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Perturbations of eigenvalues of non-normal matrices

By A. van der Sluis

Communications of the ACM, Vol. 18 No. 1, Pages 30-36

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The problem considered is to give bounds for finite perturbations of simple and multiple eigenvalues &lgr;i of nonnormal matrices, where these bounds are in terms of the eigenvalues {&lgr;i}, the departure from normality &sgr;, and the Frobenius norm ‖ &Dgr;AF of the perturbation matrix, but not in terms of the eigensystem. The bounds which are derived are shown to be almost attainable for any set of all matrices of given {&lgr;i} and &sgr;. One conclusion is that, very roughly speaking, a simple eigenvalue &lgr;1 is perturbed by |&Dgr;&lgr;1| ≲ ‖ &Dgr;AF · ∏ (&sgr;/&thgr;j) where &thgr;j is of the order of magnitude of |&lgr;1 - &lgr;j|, the product being extended over all j where &thgr;j&sgr;.

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