# The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

## Abstract

*SIAM J. Matrix Anal. Appl.*, 36 (2015), pp. 273--288] and prove that the preconditioned matrix-sequence is distributed in the eigenvalue sense as $\phi_1$ under the mild assumption that $f$ is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.

### Keywords

### MSC codes

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*On bilinear forms in Gaussian random variables and Toeplitz matrices*, Probab. Theory Related Fields, 79 (1988), pp. 37--45.

*An Introduction to Iterative Toeplitz Solvers*, Fundam. Algorithms 5, SIAM, Philadelphia, PA, 2007.

*Superoptimal approximation for unbounded symbols*, Linear Algebra Appl., 428 (2008), pp. 564--585.

*Spectral clustering properties of block multilevel Hankel matrices*, Linear Algebra Appl., 306 (2000), pp. 155--163.

*Generalized Locally Toeplitz Sequences: Theory and Applications,*Vol. I, Springer, Cham, 2017.

*A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences*, Oper. Matrices, 9 (2015), pp. 549--561.

*Toeplitz Forms and Their Applications*, 2nd ed., Chelsea, New York, 1984.

*Circulant preconditioners for functions of Hermitian Toeplitz matrices*, J. Comput. Appl. Math., 352 (2019), pp. 328--340.

*Optimal preconditioners for systems defined by functions of Toeplitz matrices*, Linear Algebra Appl., 548 (2018), pp. 148--171.

*A note on the spectral distribution of symmetrized Toeplitz sequences*, Linear Algebra Appl., 579 (2019), pp. 32--50.

*Circulant preconditioners for analytic functions of Toeplitz matrices*, Numer. Algorithms, 79 (2018), pp. 1211--1230.

*Spectral properties of flipped Toeplitz matrices*, BIT, 59 (2019), pp. 463--482.

*Iterative Methods for Toeplitz Systems*, Numer. Math. Sci. Comput., Oxford University Press, New York, 2004.

*On the distribution of the singular values of Toeplitz matrices*, Linear Algebra Appl., 80 (1986), pp. 115--130.

*A preconditioned MINRES method for nonsymmetric Toeplitz matrices*, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 273--288.

*Korovkin tests, approximation, and ergodic theory*, Math. Comp., 69 (2000), pp. 1533--1558.

*Distribution results on the algebra generated by Toeplitz sequences: A finite-dimensional approach*, Linear Algebra Appl., 328 (2001), pp. 121--130.

*Generalized locally Toeplitz sequences: Spectral analysis and applications to discretized partial differential equations*, Linear Algebra Appl., 366 (2003), pp. 371--402.

*The GLT class as a generalized Fourier analysis and applications*, Linear Algebra Appl., 419 (2006), pp. 180--233.

*A note on the spectral distribution of Toeplitz matrices*, Linear Multilinear Algebra, 45 (1998), pp. 147--159.

*New theorems on the distribution of eigenvalues and singular values of multilevel Toeplitz matrices*, Dokl. Akad. Nauk, 333 (1993), pp. 300--303.

*Influence of matrix operations on the distribution of eigenvalues and singular values of Toeplitz matrices*, Linear Algebra Appl., 207 (1994), pp. 225--249.

*A unifying approach to some old and new theorems on distribution and clustering*, Linear Algebra Appl., 232 (1996), pp. 1--43.

*Distribution of the eigenvalues and singular numbers of Toeplitz matrices under weakened requirements on the generating function*, Mat. Sb., 188 (1997), pp. 83--92.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 14 August 2018

**Accepted**: 8 July 2019

**Published online**: 12 September 2019

#### Keywords

#### MSC codes

### Authors

#### Funding Information

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- A Preconditioned MINRES Method for Optimal Control of Wave Equations and its Asymptotic Spectral Distribution TheorySIAM Journal on Matrix Analysis and Applications, Vol. 44, No. 4 | 16 October 2023
- New approach on the study of operator matrixGeorgian Mathematical Journal, Vol. 0, No. 0 | 4 October 2023
- A sine transform based preconditioned MINRES method for all-at-once systems from constant and variable-coefficient evolutionary PDEsNumerical Algorithms, Vol. 39 | 14 August 2023
- An Accelerated Iterative Method to Find the Sign of a Nonsingular Matrix with Quartical ConvergenceIranian Journal of Science, Vol. 47, No. 4 | 9 August 2023
- Generalized Drazin Invertibility of Operator MatricesNumerical Functional Analysis and Optimization, Vol. 43, No. 16 | 28 October 2022
- Band-times-circulant preconditioners for non-symmetric Toeplitz systemsBIT Numerical Mathematics, Vol. 62, No. 2 | 4 July 2021
- Band-Toeplitz preconditioners for ill-conditioned Toeplitz systemsBIT Numerical Mathematics, Vol. 62, No. 2 | 9 August 2021
- Spectral distribution of symmetrized circulant matricesCanadian Mathematical Bulletin, Vol. 65, No. 2 | 3 June 2021
- Optimal block circulant preconditioners for block Toeplitz systems with application to evolutionary PDEsJournal of Computational and Applied Mathematics, Vol. 407 | 1 Jun 2022
- Spectral distribution of families of Hankel matricesLinear Algebra and its Applications, Vol. 624 | 1 Sep 2021
- Band Preconditioners for Non-Symmetric Real Toeplitz Systems with Unknown Generating Function2021 25th International Conference on Circuits, Systems, Communications and Computers (CSCC) | 1 Jul 2021
- The Asymptotic Spectrum of Flipped Multilevel Toeplitz Matrices and of Certain PreconditioningsSIAM Journal on Matrix Analysis and Applications, Vol. 42, No. 3 | 26 August 2021
- Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioningNumerical Linear Algebra with Applications, Vol. 28, No. 1 | 15 September 2020
- Preconditioners for Krylov subspace methods: An overviewGAMM-Mitteilungen, Vol. 43, No. 4 | 21 October 2020
- Spectral properties of flipped Toeplitz matrices and related preconditioningBIT Numerical Mathematics, Vol. 59, No. 2 | 18 December 2018
- Numerical Investigation of the Spectral Distribution of Toeplitz-Function SequencesComputational Methods for Inverse Problems in Imaging | 27 November 2019