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Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. (English) Zbl 0748.15010
The \(\varepsilon\)-pseudo-eigenvalues of an \(N\times N\) matrix \(A\) are those complex numbers \(z\) for which \(\|(zI-A)^{-1}\|_ 2\geq 1/\varepsilon>0\). The authors’ results, which are partly empirical, show that, for small \(\varepsilon\) and large \(N\), the \(\varepsilon\)- pseudospectrum of a Toeplitz matrix is roughly the same as the spectrum of the associated Toeplitz operator. The corresponding pseudo- eigenvectors are also investigated.
The authors argue that the existing very different results on the exact spectra of nonnormal Toeplitz matrices are of dubious practical significance.

15A18 Eigenvalues, singular values, and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
15A42 Inequalities involving eigenvalues and eigenvectors
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