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The inverse eigenvalue problem for real symmetric Toeplitz matrices


Author: H. J. Landau
Journal: J. Amer. Math. Soc. 7 (1994), 749-767
MSC: Primary 15A18; Secondary 15A21, 15A60, 47B35
MathSciNet review: 1234570
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Abstract: We show that every set of $ n$ real numbers is the set of eigenvalues of an $ n \times n$ real symmetric Toeplitz matrix; the matrix has a certain additional regularity. The argument--based on the topological degree--is nonconstructive.


References [Enhancements On Off] (What's this?)

  • [A] N. I. Akhiezer, The classical moment problem, Hafner, New York, 1965.
  • [B] M. Berger and M. Berger, Perspectives in nonlinearity, Benjamin, New York, 1968.
  • [DG] P. Delsarte and Y. Genin, Spectral properties of finite Toeplitz matrices, Mathematical theory of networks and systems (Beer Sheva, 1983) Lecture Notes in Control and Inform. Sci., vol. 58, Springer, London, 1984, pp. 194–213. MR 792106, 10.1007/BFb0031053
  • [F] Shmuel Friedland, Inverse eigenvalue problems for symmetric Toeplitz matrices, SIAM J. Matrix Anal. Appl. 13 (1992), no. 4, 1142–1153. MR 1182718, 10.1137/0613069
  • [I] I. S. Iohvidov, Hankel and Toeplitz matrices and forms, Birkhäuser, Boston, Mass., 1982. Algebraic theory; Translated from the Russian by G. Philip A. Thijsse; With an introduction by I. Gohberg. MR 677503
  • [K] T. Kato, Perturbation theory for linear operators, Springer, New York, 1984.
  • [RN] F. Riesz and B. Sz-Nagy, Functional analysis, Ungar, New York, 1971.
  • [S] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR 0433481
  • [SL] David Slepian and Henry J. Landau, A note on the eigenvalues of Hermitian matrices, SIAM J. Math. Anal. 9 (1978), no. 2, 291–297. MR 0466171
  • [T1] William F. Trench, Spectral evolution of a one-parameter extension of a real symmetric Toeplitz matrix, SIAM J. Matrix Anal. Appl. 11 (1990), no. 4, 601–611. MR 1066162, 10.1137/0611043
  • [T2] William F. Trench, Interlacement of the even and odd spectra of real symmetric Toeplitz matrices, Linear Algebra Appl. 195 (1993), 59–68. MR 1253269, 10.1016/0024-3795(93)90256-N

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Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1994-1234570-6
Keywords: Toeplitz matrices, inverse eigenvalue problem, regular Toeplitz matrix
Article copyright: © Copyright 1994 American Mathematical Society