Asymptotics of eigenvalues for Toeplitz matrices with rational symbols that have a minimum of the 4th order

2021 ◽  
pp. 1-25
Author(s):  
Mauricio Barrera ◽  
Sergei M. Grudsky
Keyword(s):  
Toeplitz Matrices ◽  
Asymptotics Of Eigenvalues ◽  
Rational Symbols

Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

2017 ◽  
Vol 208 (11) ◽  
pp. 1578-1601 ◽  
Author(s):  
A. Böttcher ◽  
J. M. Bogoya ◽  
S. M. Grudsky ◽  
E. A. Maximenko
Keyword(s):  
Toeplitz Matrices ◽  
Eigenvalues And Eigenvectors ◽  
Asymptotics Of Eigenvalues

scholarly journals Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols

2019 ◽  
Vol 580 ◽  
pp. 292-335
Author(s):  
A.A. Batalshchikov ◽  
S.M. Grudsky ◽  
I.S. Malisheva ◽  
S.S. Mihalkovich ◽  
E. Ramírez de Arellano ◽  
...  
Keyword(s):  
Toeplitz Matrices ◽  
Asymptotics Of Eigenvalues ◽  
Simple Loop

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

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.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

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