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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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[1] Adamjan, V. M.; Arov, D. Z.; Krein, M. G., Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Math. USSR—Sb., 15, 31-73 (1971) · Zbl 0248.47019
[2] Anselone, P. M.; Sloan, I. H., Spectral approximations for Wiener-Hopf operators, J. Integral Equations Appl., 2, 237-261 (1990) · Zbl 0706.45002
[3] Bakhvalov, N. S., Numerical Methods (1977), Mir: Mir Moscow · Zbl 0766.65128
[4] Calderón, A.; Spitzer, F.; Widom, H., Inversion of Toeplitz matrices, Illinois J. Math., 3, 490-498 (1959) · Zbl 0091.11101
[5] Chan, R. H.; Strang, G., Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Statist. Comput., 10, 104-119 (1989) · Zbl 0666.65030
[6] Day, K. M., Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions, Trans. Amer. Math. Soc., 209, 175-183 (1975) · Zbl 0324.47017
[7] Devinatz, A., Toeplitz operators on \(H^2\) spaces, Trans. Amer. Math. Soc., 112, 304-317 (1964) · Zbl 0139.07202
[8] Edelman, A., Eigenvalues and Condition Numbers of Random Matrices, (Ph.D. Dissertation (1989), Dept. of Mathematics, M.I.T) · Zbl 0678.15019
[9] Godunov, S. K.; Ryabenkii, V. S., Theory of Difference Schemes (1964), North-Holland: North-Holland Amsterdam
[10] Gohberg, I. C.; Fel’dman, I. A., Convolution Equations and Projection Methods for Their Solution (1974), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0278.45008
[11] Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), Johns Hopkins U.P: Johns Hopkins U.P Baltimore · Zbl 0733.65016
[12] Grenander, U.; Szegö, G., Toeplitz Forms and Their Applications (1958), Univ. of California Press: Univ. of California Press Berkeley · Zbl 0080.09501
[13] Halmos, P. R., A Hilbert Space Problem Book (1967), Van Nostrand: Van Nostrand New York · Zbl 0144.38704
[14] Hirschman, I. I., The spectra of certain Toeplitz matrices, Illinois J. Math., 11, 145-159 (1967) · Zbl 0144.38501
[15] Kac, M.; Murdock, W. L.; Szegö, G., On the eigen-values of certain hermitian forms, J. Rational Mech. Anal., 2, 767-800 (1953) · Zbl 0051.30302
[16] Kahan, W., Numerical linear algebra, Canadian Math. Bull., 9, 757-801 (1965) · Zbl 0236.65025
[17] Kato, T., Perturbation Theory for Linear Operators (1976), Springer-Verlag: Springer-Verlag New York
[18] Krein, M. G., Integral equations on a half-line whose kernel depends on the difference of its arguments, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Amer. Math. Soc. Transl. Ser. 2, 22, 163-288 (1962), English transl. · Zbl 0119.09601
[19] Lerer, L. E., On the asymptotic distribution of the spectra of finite truncations of Wiener-Hopf operators, Dokl. Akad. Nauk. SSSR, 207, 1651-1655 (1972) · Zbl 0268.47030
[20] N. M. Nachtigal, L. Reichel, and L. N. Trefethen, A hybrid GMRES algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. Appl.; N. M. Nachtigal, L. Reichel, and L. N. Trefethen, A hybrid GMRES algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. Appl. · Zbl 0757.65035
[21] Reddy, S. C.; Trefethen, L. N., Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues, Comput. Methods Appl. Meth. Engrg., 80, 147-164 (1990) · Zbl 0735.65070
[22] Schmidt, P.; Spitzer, F., The Toeplitz matrices of an arbitrary Laurent polynomial, Math. Scand., 8, 15-38 (1960) · Zbl 0101.09203
[23] Toeplitz, O., Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veränderlichen, Math. Ann., 70, 351-376 (1911) · JFM 42.0366.01
[24] Trefethen, L. N., Rational Chebyshev approximation on the unit disk, Numer. Math., 37, 297-320 (1981) · Zbl 0443.30046
[25] Trefethen, L. N., Approximation theory and numerical linear algebra, (Mason, J. C.; Cox, M. G., Algorithms for Approximation II (1990), Chapman: Chapman London) · Zbl 0747.41005
[26] L. N. Tefethen, Non-normal Matrices and Pseudospectra; L. N. Tefethen, Non-normal Matrices and Pseudospectra
[27] Ullman, J. L., A problem of Schmidt and Spitzer, Bull. Amer. Math. Soc., 73, 883-885 (1967) · Zbl 0167.13101
[28] Ullman, J. L., Toeplitz matrices associated with a semi-infinite Laurent series, Proc. London Math. Soc., 22, 3, 164-192 (1971) · Zbl 0208.16602
[29] Varah, J. M., On the separation of two matrices, SIAM J. Numer. Anal., 16, 215-222 (1979) · Zbl 0435.65034
[30] R. M. Beam and R. F. Warming, The asymptotic eigenvalue spectra of banded Toeplitz and quasi-Toeplitz matrices, subm. to SIAM J. Sci. Stat. Comput.; R. M. Beam and R. F. Warming, The asymptotic eigenvalue spectra of banded Toeplitz and quasi-Toeplitz matrices, subm. to SIAM J. Sci. Stat. Comput. · Zbl 0788.65049
[31] Widom, H., Toeplitz matrices, (Hirschman, I. I., Studies in Real and Complex Analysis (1965), Math. Assoc. Amer) · Zbl 0173.42501
[32] Widom, H., On the singular values of Toeplitz matrices, Z. Anal. Anwendungen, 8, 221-229 (1989) · Zbl 0692.47028
[33] Widom, H., Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanishing index, Oper. Theory: Adv. Appl., 48, 387-421 (1990)
[34] Wintner, A., Zur Theorie der beschränkten Bilinearformen, Math. Z., 30, 228-282 (1929) · JFM 55.0826.01
[35] Reddy, S. C., Pseudospectra of Operators and Discretization Matrices and an Application to Stability of the Method of Lines, (Ph.D. Dissertation (1991), Dept. of Mathematics, M.I.T)
[36] L. N. Trefethen, Pseudospectra of matrices, in D.F. Griffiths and G.A. Watson, eds., Proc. 14th Dundee Biennial Conference on Numerical Analysis; L. N. Trefethen, Pseudospectra of matrices, in D.F. Griffiths and G.A. Watson, eds., Proc. 14th Dundee Biennial Conference on Numerical Analysis · Zbl 0798.15005
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