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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.

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