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Journal articles on the topic 'Matrice de Toeplitz'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

Rambour, Philippe. "Valeur propre minimale d’une matrice de Toeplitz et d’un produit de matrices de Toeplitz." Annales mathématiques du Québec 39, no. 1 (June 2015): 25–48. http://dx.doi.org/10.1007/s40316-015-0033-7.

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2

Rambour, Philippe, Jean-Marc Rinkel, and Abdellatif Seghier. "Inverse asymptotique de la matrice de Toeplitz et noyau de Green." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 331, no. 11 (December 2000): 857–60. http://dx.doi.org/10.1016/s0764-4442(00)01735-3.

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3

Seghier, A. "Inversion de la matrice de Toeplitz en d dimensions et développement asymptotique de la trace de l'inverse à l'ordre d." Journal of Functional Analysis 67, no. 3 (July 1986): 380–412. http://dx.doi.org/10.1016/0022-1236(86)90032-7.

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4

Rambour, Philippe, and Jean-Marc Rinkel. "Un théorème de Spitzer-Stone fort pour une matrice de Toeplitz à symbole singulier défini par une classe de fonctions analytiques." Annales de la faculté des sciences de Toulouse Mathématiques 16, no. 2 (2007): 331–67. http://dx.doi.org/10.5802/afst.1151.

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5

Cao, Lei, and Selcuk Koyuncu. "A note on multilevel Toeplitz matrices." Special Matrices 7, no. 1 (January 1, 2019): 114–26. http://dx.doi.org/10.1515/spma-2019-0011.

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Abstract Chien, Liu, Nakazato and Tam proved that all n × n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.

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6

Altun, Muhammed. "Fine Spectra of Symmetric Toeplitz Operators." Abstract and Applied Analysis 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/932785.

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The fine spectra of 2-banded and 3-banded infinite Toeplitz matrices were examined by several authors. The fine spectra ofn-banded triangular Toeplitz matrices and tridiagonal symmetric matrices were computed in the following papers: Altun, “On the fine spectra of triangular toeplitz operators” (2011) and Altun, “Fine spectra of tridiagonal symmetric matrices” (2011). Here, we generalize those results to the ()-banded symmetric Toeplitz matrix operators for arbitrary positive integer .

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7

Lv, Xiao-Guang, and Ting-Zhu Huang. "The Inverses of Block Toeplitz Matrices." Journal of Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/207176.

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We study the inverses of block Toeplitz matrices based on the analysis of the block cyclic displacement. New formulas for the inverses of block Toeplitz matrices are proposed. We show that the inverses of block Toeplitz matrices can be decomposed as a sum of products of block circulant matrices. In the scalar case, the inverse formulas are proved to be numerically forward stable, if the Toeplitz matrix is nonsingular and well conditioned.

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8

Khan, Muhammad Ahsan. "A Family of Maximal Algebras of Block Toeplitz Matrices." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 3 (December 1, 2018): 127–42. http://dx.doi.org/10.2478/auom-2018-0037.

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AbstractThe maximal commutative subalgebras containing only Toeplitz matrices have been identified as generalized circulants. A similar simple description cannot be obtained for block Toeplitz matrices. We introduce and investigate certain families of maximal commutative algebras of block Toeplitz matrices.

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9

Gutiérrez-Gutiérrez, Jesús, Xabier Insausti, and Marta Zárraga-Rodríguez. "Applications of the Periodogram Method for Perturbed Block Toeplitz Matrices in Statistical Signal Processing." Mathematics 8, no. 4 (April 14, 2020): 582. http://dx.doi.org/10.3390/math8040582.

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In this paper, we combine the periodogram method for perturbed block Toeplitz matrices with the Cholesky decomposition to give a parameter estimation method for any perturbed vector autoregressive (VAR) or vector moving average (VMA) process, when we only know a perturbed version of the sequence of correlation matrices of the process. In order to combine the periodogram method for perturbed block Toeplitz matrices with the Cholesky decomposition, we first need to generalize a known result on the Cholesky decomposition of Toeplitz matrices to perturbed block Toeplitz matrices.

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10

Park, Ju Yong, Jeong Su Kim, Ferenc Szollosi, and Moon Ho Lee. "The Toeplitz Circulant Jacket Matrices." Journal of the Institute of Electronics and Information Engineers 50, no. 7 (July 25, 2013): 19–26. http://dx.doi.org/10.5573/ieek.2013.50.7.019.

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11

Jiang, Zhao-Lin, Xiao-Ting Chen, and Jian-Min Wang. "The Explicit Inverses of CUPL-Toeplitz and CUPL-Hankel Matrices." East Asian Journal on Applied Mathematics 7, no. 1 (January 31, 2017): 38–54. http://dx.doi.org/10.4208/eajam.070816.191016a.

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AbstractIn this paper, we consider two innovative structured matrices, CUPL-Toeplitz matrix and CUPL-Hankel matrix. The inverses of CUPL-Toeplitz and CUPL-Hankel matrices can be expressed by the Gohberg-Heinig type formulas, and the stability of the inverse matrices is verified in terms of 1-, ∞- and 2-norms, respectively. In addition, two algorithms for the inverses of CUPL-Toeplitz and CUPL-Hankel matrices are given and examples are provided to verify the feasibility of these algorithms.

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12

Zhang, Maoyun, Xiaoyu Jiang, and Zhaolin Jiang. "Explicit determinants, inverses and eigenvalues of four band Toeplitz matrices with perturbed rows." Special Matrices 7, no. 1 (January 1, 2019): 52–66. http://dx.doi.org/10.1515/spma-2019-0004.

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Abstract In this paper, four-band Toeplitz matrices and four-band Hankel matrices of type I and type II with perturbed rows are introduced. Explicit determinants, inverses and eigenvalues for these matrices are derived by using a nice inverse formula of block bidiagonal Toeplitz matrices.

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13

Collao, Macarena, Mario Salas, and Ricardo L. Soto. "Toeplitz nonnegative realization of spectra via companion matrices." Special Matrices 7, no. 1 (January 1, 2019): 230–45. http://dx.doi.org/10.1515/spma-2019-0017.

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Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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14

Shi, Baijuan. "On the Norms ofr-Hankel andr-Toeplitz Matrices." Mathematical Problems in Engineering 2019 (February 11, 2019): 1–4. http://dx.doi.org/10.1155/2019/6729701.

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In this paper, using the properties ofr-Hankel andr-Toeplitz matrices, combining the properties of exponential form, we shall study the spectral norms ofr-Hankel andr–Toeplitz matrices involving exponential forme(x).

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15

BOSE, ARUP, SREELA GANGOPADHYAY, and KOUSHIK SAHA. "CONVERGENCE OF A CLASS OF TOEPLITZ TYPE MATRICES." Random Matrices: Theory and Applications 02, no. 03 (July 2013): 1350006. http://dx.doi.org/10.1142/s2010326313500068.

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We use the method of moments to study the spectral properties in the bulk for finite diagonal large dimensional random and non-random Toeplitz type matrices via the joint convergence of matrices in an appropriate sense. As a consequence we revisit the famous limit theorem of Szegö for non-random symmetric Toeplitz matrices.

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16

Stanimirović, Predrag, Marko Miladinović, Igor Stojanović, and Sladjana Miljković. "Application of the partitioning method to specific Toeplitz matrices." International Journal of Applied Mathematics and Computer Science 23, no. 4 (December 1, 2013): 809–21. http://dx.doi.org/10.2478/amcs-2013-0061.

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Abstract We propose an adaptation of the partitioning method for determination of theMoore-Penrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant reduction in the computational time required to calculate the Moore-Penrose inverse of specific Toeplitz matrices of an arbitrary size. The method is implemented in MATLAB, and illustrative examples are presented.

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17

Maddax, Ivor J. "GENERALIZED TOEPLITZ MATRICES." Analysis 12, no. 3-4 (December 1992): 335–42. http://dx.doi.org/10.1524/anly.1992.12.34.335.

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18

Farenick, Douglas R., Mark Krupnik, Naum Krupnik, and Woo Young Lee. "Normal Toeplitz Matrices." SIAM Journal on Matrix Analysis and Applications 17, no. 4 (October 1996): 1037–43. http://dx.doi.org/10.1137/s0895479895287080.

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19

Maddox, I. J. "Uniform Toeplitz matrices." International Journal of Mathematics and Mathematical Sciences 13, no. 2 (1990): 227–32. http://dx.doi.org/10.1155/s0161171290000345.

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We characterize all infinite matrices of bounded linear operators on a Banach space which preserve the limits of uniformly convergent sequences defined on an infinite set. Also, we give a Tauberian theorem for uniform summability by the Kuttner-Maddox matrix.

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20

Filipiak, Katarzyna, Augustyn Markiewicz, Adam Mieldzioc, and Aneta Sawikowska. "On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices." Electronic Journal of Linear Algebra 33 (May 16, 2018): 74–82. http://dx.doi.org/10.13001/1081-3810.3750.

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We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.

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21

Fu, Yaru, Xiaoyu Jiang, Zhaolin Jiang, and Seongtae Jhang. "Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns." Special Matrices 8, no. 1 (May 4, 2020): 131–43. http://dx.doi.org/10.1515/spma-2020-0012.

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AbstractIn this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) {\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

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22

Zhang, Weiqian, and Chaoqian Li. "An Eigenvalue Inclusion Set for Matrices with a Constant Main Diagonal Entry." Symmetry 10, no. 12 (December 12, 2018): 745. http://dx.doi.org/10.3390/sym10120745.

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A set to locate all eigenvalues for matrices with a constant main diagonal entry is given, and it is proved that this set is tighter than the well-known Geršgorin set, the Brauer set and the set proposed in (Linear and Multilinear Algebra, 60:189-199, 2012). Furthermore, by applying this result to Toeplitz matrices as a subclass of matrices with a constant main diagonal, we obtain a set including all eigenvalues of Toeplitz matrices.

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23

Il'in, S. N. "Infinite Limitedly-Toeplitz Extensions of Toeplitz Matrices." Journal of Mathematical Sciences 132, no. 2 (January 2006): 160–65. http://dx.doi.org/10.1007/s10958-005-0486-3.

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24

Zimmerman, Dale L. "Block toeplitz products of block toeplitz matrices." Linear and Multilinear Algebra 25, no. 3 (October 1989): 185–90. http://dx.doi.org/10.1080/03081088908817940.

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25

Zhang, Rui, Chen Meng, Cheng Wang, and Qiang Wang. "A Novel Optimization Method for Bipolar Chaotic Toeplitz Measurement Matrix in Compressed Sensing." Journal of Sensors 2021 (July 30, 2021): 1–11. http://dx.doi.org/10.1155/2021/4024737.

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In this paper, a bipolar chaotic Toeplitz measurement matrix optimization algorithm for alternating optimization is presented. The construction of measurement matrices is one of the key techniques for compressive sensing from theory to engineering applications. Recent studies have shown that bipolar chaotic Toeplitz matrices, constructed by combining the intrinsic determinism of bipolar chaotic sequences with the advantages of Toeplitz matrices, have significant advantages over other measurement matrices in terms of memory overhead, computational complexity, and hard implementation. However, problems such as strong correlation and large interdependence coefficients between measurement matrices and sparse dictionaries may still exist in practical applications. To address this problem, we propose a new bipolar chaotic Toeplitz measurement matrix alternating optimization algorithm. Firstly, by introducing the structure matrix, the optimization problem of the measurement matrix is transformed into the optimization problem of the generating sequence, thus ensuring that the optimization process does not destroy the structural properties of the matrix; then, constraints are added to the values of the generating sequence during the optimization process, so that the optimized measurement matrix still maintains the bipolar properties. Finally, the effectiveness of the optimization algorithm in this paper is verified by simulation experiments. The experimental results show that the optimized bipolar chaotic Toeplitz measurement matrix can effectively reduce the reconstruction error and improve the reconstruction probability.

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26

Tian, Li Chao, and Hong Kui Li. "An Improved Fast Algorithm for Solving Toeplitz Systems in Mechanical Control." Applied Mechanics and Materials 55-57 (May 2011): 863–67. http://dx.doi.org/10.4028/www.scientific.net/amm.55-57.863.

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Pentadiagonal Toeplitz systems of linear equations arise in many application areas. Because of the structure and many good properties of pentadiagonal Toeplitz matrices, they have been applied in Mechanical Control. Based on [1], in this paper, we present an improved fast algorithm for solving symmetric pentadiagonal Toeplitz systems.

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27

Gover, M. J. C. "The determination of companion matrices characterizing toeplitz and r-Toeplitz matrices." Linear Algebra and its Applications 117 (May 1989): 81–92. http://dx.doi.org/10.1016/0024-3795(89)90549-1.

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28

Miller, Steven J., Kirk Swanson, Kimsy Tor, and Karl Winsor. "Limiting spectral measures for random matrix ensembles with a polynomial link function." Random Matrices: Theory and Applications 04, no. 02 (April 2015): 1550004. http://dx.doi.org/10.1142/s2010326315500045.

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Consider the ensembles of real symmetric Toeplitz matrices and real symmetric Hankel matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work on real symmetric Toeplitz matrices shows that the spectral measures, or densities of normalized eigenvalues, converge almost surely to a universal near-Gaussian distribution, while previous work on real symmetric Hankel matrices shows that the spectral measures converge almost surely to a universal non-unimodal distribution. Real symmetric Toeplitz matrices are constant along the diagonals, while real symmetric Hankel matrices are constant along the skew diagonals. We generalize the Toeplitz and Hankel matrices to study matrices that are constant along some curve described by a real-valued bivariate polynomial (other authors refer to the dependencies among the matrix elements as arising from a link function). Using the Method of Moments and an analysis of the resulting Diophantine equations, we show that the spectral measures associated with linear bivariate polynomials converge in probability and almost surely to universal non-semicircular distributions. We prove that for certain choices these limiting distributions approach the semicircle in the limit of large values of the polynomial coefficients. We then prove that the spectral measures associated with the sum or difference of any two real-valued polynomials with different degrees converge in probability and almost surely to a universal semicircular distribution.

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29

Perline, Ronald. "Toeplitz Matrices and Commuting Tridiagonal Matrices." SIAM Journal on Matrix Analysis and Applications 12, no. 2 (April 1991): 321–26. http://dx.doi.org/10.1137/0612023.

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30

Nobari, Elham, and Bijan Ahmadi Kakavandi. "A geometric mean for Toeplitz and Toeplitz-block block-Toeplitz matrices." Linear Algebra and its Applications 548 (July 2018): 189–202. http://dx.doi.org/10.1016/j.laa.2018.03.014.

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31

Pestana, J. "Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices." SIAM Journal on Matrix Analysis and Applications 40, no. 3 (January 2019): 870–87. http://dx.doi.org/10.1137/18m1205406.

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32

Hanke, Martin, and James G. Nagy. "Toeplitz approximate inverse preconditioner for banded Toeplitz matrices." Numerical Algorithms 7, no. 2 (September 1994): 183–99. http://dx.doi.org/10.1007/bf02140682.

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33

Mildenberger, Heike. "Toeplitz matrices and convergence." Fundamenta Mathematicae 165, no. 2 (2000): 175–89. http://dx.doi.org/10.4064/fm-165-2-175-189.

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34

Babenko, K. I. "Toeplitz and Hankel matrices." Russian Mathematical Surveys 41, no. 1 (February 28, 1986): 209–18. http://dx.doi.org/10.1070/rm1986v041n01abeh003212.

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35

Böttcher, Albrecht. "Orthogonal Symmetric Toeplitz Matrices." Complex Analysis and Operator Theory 2, no. 2 (March 17, 2008): 285–98. http://dx.doi.org/10.1007/s11785-008-0053-2.

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36

Zhang, Er Yan, and Xiao Feng Zhu. "The Recursive Algorithms of Yule-Walker Equation in Generalized Stationary Prediction." Advanced Materials Research 756-759 (September 2013): 3070–73. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.3070.

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Toeplitz matrix arises in a remarkable variety of applications such as signal processing, time series analysis, image processing. Yule-Walker equation in generalized stationary prediction is linear algebraic equations that use Toeplitz matrix as coefficient matrix. Making better use of the structure of Toeplitz matrix, we present a recursive algorithm of linear algebraic equations from by using Toeplitz matrix as coefficient matrix , and also offer the proof of the recursive formula. The algorithm, making better use of the structure of Toeplitz matrices, effectively reduces calculation cost. For n-order Toeplitz coefficient matrix, the computational complexity of usual Gaussian elimination is about , while this algorithm is about , decreasing of one order of magnitude.

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37

Banerjee, Debapratim, and Arup Bose. "Patterned sparse random matrices: A moment approach." Random Matrices: Theory and Applications 06, no. 03 (July 2017): 1750011. http://dx.doi.org/10.1142/s2010326317500113.

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We consider four specific [Formula: see text] sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability [Formula: see text] such that [Formula: see text] with [Formula: see text]. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

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38

Radhika, Varadharajan, Jay M. Jahangiri, Srikandan Sivasubramanian, and Gangadharan Murugusundaramoorthy. "Toeplitz matrices whose elements are coefficients of Bazilevič functions." Open Mathematics 16, no. 1 (October 29, 2018): 1161–69. http://dx.doi.org/10.1515/math-2018-0093.

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AbstractWe consider the Toeplitz matrices whose elements are the coefficients of Bazilevič functions and obtain upper bounds for the first four determinants of these Toeplitz matrices. The results presented here are new and noble and the only prior compatible results are the recent publications by Thomas and Halim [1] for the classes of starlike and close-to-convex functions and Radhika et al. [2] for the class of functions with bounded boundary rotation.

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39

Verde-Star, Luis. "Polynomial sequences generated by infinite Hessenberg matrices." Special Matrices 5, no. 1 (January 26, 2017): 64–72. http://dx.doi.org/10.1515/spma-2017-0002.

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Abstract We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

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40

Kailath, T., and J. Chun. "Generalized Displacement Structure for Block-Toeplitz, Toeplitz-Block, and Toeplitz-Derived Matrices." SIAM Journal on Matrix Analysis and Applications 15, no. 1 (January 1994): 114–28. http://dx.doi.org/10.1137/s0895479889169042.

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41

Deng, Quanling. "Analytical solutions to some generalized and polynomial eigenvalue problems." Special Matrices 9, no. 1 (January 1, 2021): 240–56. http://dx.doi.org/10.1515/spma-2020-0135.

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Abstract It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.

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42

YALÇINER, AYNUR. "THE LU FACTORIZATIONS AND DETERMINANTS OF THE K-TRIDIAGONAL MATRICES." Asian-European Journal of Mathematics 04, no. 01 (March 2011): 187–97. http://dx.doi.org/10.1142/s1793557111000162.

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43

Ren, Zhi-Ru. "Banded Toeplitz Preconditioners for Toeplitz Matrices from Sinc Methods." Journal of Computational Mathematics 30, no. 5 (June 2012): 533–43. http://dx.doi.org/10.4208/jcm.1203-m3761.

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44

Griffin, Kent, Jeffrey L. Stuart, and Michael J. Tsatsomeros. "Noncirculant Toeplitz matrices all of whose powers are Toeplitz." Czechoslovak Mathematical Journal 58, no. 4 (December 2008): 1185–93. http://dx.doi.org/10.1007/s10587-008-0078-8.

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45

Gallivan, K. A., S. Thirumalai, P. Van Dooren, and V. Vermaut. "High performance algorithms for Toeplitz and block Toeplitz matrices." Linear Algebra and its Applications 241-243 (July 1996): 343–88. http://dx.doi.org/10.1016/0024-3795(95)00649-4.

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Ben-Artzi, Asher, and Tamir Shalom. "On inversion of Toeplitz and close to Toeplitz matrices." Linear Algebra and its Applications 75 (March 1986): 173–92. http://dx.doi.org/10.1016/0024-3795(86)90188-6.

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Gao, Yuefeng, Jianlong Chen, Pedro Patrício, and Dingguo Wang. "The pseudo core inverse of a companion matrix." Studia Scientiarum Mathematicarum Hungarica 55, no. 3 (September 2018): 407–20. http://dx.doi.org/10.1556/012.2018.55.3.1398.

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Abstract:

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1,3}-inverse of a Toeplitz matrix plays an important role in that process.

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Rost, Karla. "Toeplitz-plus-Hankel Bezoutians and inverses of Toeplitz and Toeplitz-plus-Hankel matrices." Operators and Matrices, no. 3 (2008): 385–406. http://dx.doi.org/10.7153/oam-02-23.

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Hilberdink, Titus. "Determinants of multiplicative Toeplitz matrices." Acta Arithmetica 125, no. 3 (2006): 265–84. http://dx.doi.org/10.4064/aa125-3-4.

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Evans, Ron, and Nolan Wallach. "Pfaffians of Toeplitz payoff matrices." Linear Algebra and its Applications 577 (September 2019): 114–20. http://dx.doi.org/10.1016/j.laa.2019.04.026.

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