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Eugene Bogomolny - One of the best experts on this subject based on the ideXlab platform.

  • Spectral statistics of random Toeplitz Matrices
    Physical Review E, 2020
    Co-Authors: Eugene Bogomolny
    Abstract:

    Spectral statistics of hermitian random Toeplitz Matrices with independent identically distributed elements is investigated numerically. It is found that the eigenvalue statistics of complex Toeplitz Matrices is surprisingly well approximated by the semi-Poisson distribution belonging to intermediate-type statistics observed in certain pseudo-integrable billiards. The origin of intermediate behaviour could be attributed to the fact that Fourier transformed random Toeplitz Matrices have the same slow decay outside the main diagonal as critical random matrix ensembles. The statistical properties of the full spectrum of real random Toeplitz Matrices with i.i.d. elements are close to the Poisson distribution but each of their constituted sub-spectra is again well described by the semi-Poisson distribution. The findings open new perspective in intermediate statistics.

  • Spectral statistics of Toeplitz Matrices
    arXiv: Quantum Physics, 2020
    Co-Authors: Eugene Bogomolny
    Abstract:

    Spectral statistics of hermitian random Toeplitz Matrices with independent identically distributed elements is investigated numerically. It is found that the eigenvalue statistics of complex Toeplitz Matrices is surprisingly well approximated by the semi-Poisson distribution belonging to intermediate-type statistics observed in certain pseudo-integrable billiards. The origin of intermediate behaviour could be attributed to the fact that Fourier transformed random Toeplitz Matrices have the same slow decay outside the main diagonal as critical random matrix ensembles. The statistical properties of the full spectrum of real random Toeplitz Matrices with i.i.d. elements are close to the Poisson distribution but each of their constituted sub-spectra is again well described by the semi-Poisson distribution. The findings open new perspective in intermediate statistics.

  • Spectral statistics of random Toeplitz Matrices.
    Physical review. E, 2020
    Co-Authors: Eugene Bogomolny
    Abstract:

    The spectral statistics of Hermitian random Toeplitz Matrices with independent and identically distributed elements are investigated numerically. It is found that eigenvalue statistics of complex Toeplitz Matrices are surprisingly well approximated by the semi-Poisson distribution belonging to intermediate-type statistics observed in certain pseudointegrable billiards. The origin of intermediate behavior could be attributed to the fact that Fourier transformed random Toeplitz Matrices have the same slow decay outside the main diagonal as critical random matrix ensembles. The statistical properties of the full spectrum of real random Toeplitz Matrices are close to the Poisson distribution, but each of their constituent subspectra is again well described by the semi-Poisson distribution. The findings indicate that intermediate statistics in general and the semi-Poisson distribution in particular are more universal than considered before.

Albrecht Böttcher - One of the best experts on this subject based on the ideXlab platform.

  • Orthogonal Symmetric Toeplitz Matrices
    Complex Analysis and Operator Theory, 2008
    Co-Authors: Albrecht Böttcher
    Abstract:

    We show that the number of orthogonal and symmetric Toeplitz Matrices of a given order is finite and determine all these Matrices. In this way we also obtain a description of the set of all symmetric Toeplitz Matrices whose spectrum is a prescribed doubleton.

  • Asymptotically Good Pseudomodes for Toeplitz Matrices and Wiener-Hopf Operators
    Operator Theoretical Methods and Applications to Mathematical Physics, 2004
    Co-Authors: Albrecht Böttcher, Sergei M. Grudsky
    Abstract:

    We describe the structure of asymptotically good pseudomodes for Toeplitz Matrices and their circulant analogues as well as for Wiener-Hopf integral operators and a continuous analogue of banded circulant Matrices. The pseudomodes of circulant Matrices and their continuous analogues are extended, while those of Toeplitz Matrices or Wiener-Hopf operators are typically strongly localized in the endpoints.

  • Piecewise Continuous Toeplitz Matrices and Operators: Slow Approach to Infinity
    SIAM Journal on Matrix Analysis and Applications, 2002
    Co-Authors: Albrecht Böttcher, Mark Embree, Lloyd N. Trefethen
    Abstract:

    The pseudospectra of banded finite dimensional Toeplitz Matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz Matrices---not just eigenvalues. What if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite Matrices still converge, but it is shown here that the rate is no longer exponential in the matrix dimension---only algebraic.

  • Toeplitz Matrices asymptotic linear algebra and functional analysis
    2000
    Co-Authors: Albrecht Böttcher, Sergei M. Grudsky
    Abstract:

    This text is a self-contained introduction to some problems for Toeplitz Matrices that are placed in the borderland between linear algebra and functional analysis. The text looks at Toeplitz Matrices with rational symbols, and focuses attention on the asymptotic behavior of the singular values, which includes the behavior of the norms, the norms of the inverses, and the condition numbers as special cases. The text illustrates that the asymptotics of several linear algebra characteristics depend in a fascinating way on functional analytic properties of infinite Matrices. Many convergence results can very comfortably be obtained by working with appropriate C*-algebras, while refinements of these results, for example, estimates of the convergence speed, nevertheless require hard analysis.

  • introduction to large truncated Toeplitz Matrices
    1998
    Co-Authors: Albrecht Böttcher, Bernd Silbermann
    Abstract:

    1 Infinite Matrices.- 1.1 Boundedness and Invertibility.- 1.2 Laurent Matrices.- 1.3 Toeplitz Matrices.- 1.4 Hankel Matrices.- 1.5 Wiener-Hopf Factorization.- 1.6 Continuous Symbols.- 1.7 Locally Sectorial Symbols.- 1.8 Discontinuous Symbols.- 2 Finite Section Method and Stability.- 2.1 Approximation Methods.- 2.2 Continuous Symbols.- 2.3 Asymptotic Inverses.- 2.4 The Gohberg-Feldman Approach.- 2.5 Algebraization of Stability.- 2.6 Local Principles.- 2.7 Localization of Stability.- 3 Norms of Inverses and Pseudospectra.- 3.1C*-Algebras.- 3.2 Continuous Symbols.- 3.3 Piecewise Continuous Symbols.- 3.4 Norm of the Resolvent.- 3.5 Limits of Pseudospectra.- 3.6 Pseudospectra of Infinite Toeplitz Matrices.- 4 Moore-Penrose Inverses and Singular Values.- 4.1 Singular Values of Matrices.- 4.2 The Lowest Singular Value.- 4.3 The Splitting Phenomenon.- 4.4 Upper Singular Values.- 4.5 Moler's Phenomenon.- 4.6 Limiting Sets of Singular Values.- 4.7 The Moore-Penrose Inverse.- 4.8 Asymptotic Moore-Penrose Inversion.- 4.9 Moore-Penrose Sequences.- 4.10 Exact Moore-Penrose Sequences.- 4.11 Regularization and Kato Numbers.- 5 Determinants and Eigenvalues.- 5.1 The Strong Szegoe Limit Theorem.- 5.2 Ising Model and Onsager Formula.- 5.3 Second-Order Trace Formulas.- 5.4 The First Szegoe Limit Theorem.- 5.5 Hermitian Toeplitz Matrices.- 5.6 The Avram-Parter Theorem.- 5.7 The Algebraic Approach to Trace Formulas.- 5.8 Toeplitz Band Matrices.- 5.9 Rational Symbols.- 5.10 Continuous Symbols.- 5.11 Fisher-Hartwig Determinants.- 5.12 Piecewise Continuous Symbols.- 6 Block Toeplitz Matrices.- 6.1 Infinite Matrices.- 6.2 Finite Section Method and Stability.- 6.3 Norms of Inverses and Pseudospectra.- 6.4 Distribution of Singular Values.- 6.5 Asymptotic Moore-Penrose Inversion.- 6.6 Trace Formulas.- 6.7 The Szegoe-Widom Limit Theorem.- 6.8 Rational Matrix Symbols.- 6.9 Multilevel Toeplitz Matrices.- 7 Banach Space Phenomena.- 7.1 Boundedness.- 7.2 Fredholmness and Invertibility.- 7.3 Continuous Symbols.- 7.4 Piecewise Continuous Symbols.- 7.5 Loss of Symmetry.- References.- Symbol Index.

Maryam Shams Solary - One of the best experts on this subject based on the ideXlab platform.

Robert F. Warming - One of the best experts on this subject based on the ideXlab platform.

Richard M. Beam - One of the best experts on this subject based on the ideXlab platform.