Tridiagonal Toeplitz Matrices Research Articles

Reversibility of linear cellular automata with intermediate boundary condition

<abstract><p>This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over $ \mathbb{Z}_2 $.</p></abstract>

On the efficient and accurate determinant evaluation of periodic tridiagonal Toeplitz matrices

On the efficient and accurate determinant evaluation of periodic tridiagonal Toeplitz matrices

Digital Uncoupling of Coupled Multi-Beam Arrays

Antenna arrays consisting of up to thousands of elements are required in a plethora of applications, such as wireless communications, microwave imaging, radio astronomy, and radar. The electromagnetic fields surrounding the antenna elements are mutually coupled; this means that an excitation of a given element also causes voltages to be induced in the terminals of the neighboring elements. Such coupling causes deviation of the expected antenna array beam response unless mitigation of the mutually-coupled fields available within the beamforming scheme. Antenna mutual coupling also causes LNA noise coupling, which can severely degenerate the noise performance of an array receiver. The mutual coupling between elements can be quantified by measuring the scattering parameters across the array using a vector network analyzer. This paper proposes a low-complexity and real-time capable algorithm that will furnish the uncoupling of mutually coupled elements in the digital signal processing back-end using a fast inversion algorithm for tridiagonal toeplitz matrices.

LCD codes from tridiagonal Toeplitz matrices

Double Toeplitz (DT) codes are codes with a generator matrix of the form (I,T) with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields.

Characterizations and accurate computations for tridiagonal Toeplitz matrices

ABSTRACT Tridiagonal Toeplitz P-matrices, M-matrices and totally positive matrices are characterized. For some classes of tridiagonal matrices and tridiagonal Toeplitz matrices, it is shown that many algebraic computations can be performed with high relative accuracy.

Nonconcavity of the spectral radius in Levinger's theorem

Let A∈Rn×n be a nonnegative irreducible square matrix and let r(A) be its spectral radius and Perron-Frobenius eigenvalue. Levinger asserted and several have proven that r(t):=r((1−t)A+tA⊤) increases over t∈[0,1/2] and decreases over t∈[1/2,1]. It has further been stated that r(t) is concave over t∈(0,1). Here we show that the latter claim is false in general through a number of counterexamples, but prove it is true for A∈R2×2, weighted shift matrices (but not cyclic weighted shift matrices), tridiagonal Toeplitz matrices, and the 3-parameter Toeplitz matrices from Fiedler, but not Toeplitz matrices in general. A general characterization of the range of t, or the class of matrices, for which the spectral radius is concave in Levinger's homotopy remains an open problem.

ON INVERSES AND EIGENPAIRS OF PERIODIC TRIDIAGONAL TOEPLITZ MATRICES WITH PERTURBED CORNERS

In this paper, we derive explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type $I$. The Mersenne numbers play an important role in these explicit formulas derived. Our main approaches include clever uses of the Schur complement and matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, the properties of Type $II$ matrix can be also obtained, which benefits from the relation between Type $I$ and $II$ matrices. Lastly, we give three algorithms for these basic quantities and analyze them to illustrate our theoretical results.

Fine Spectra of Tridiagonal Toeplitz Matrices

The fine spectra of n-banded triangular Toeplitz matrices and (2n+1)-banded symmetric Toeplitz matrices were computed in (M. Altun, Appl. Math. Comput., 217, 8044–8051 (2011)) and (M. Altun, Abstr. Appl. Anal., Article ID 932785 (2012)). As a continuation of these results, we compute the fine spectra of tridiagonal Toeplitz matrices. These matrices are, in general, neither triangular, nor symmetric.

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

In this paper, we deal mainly with a class of periodic tridiagonal Toeplitz matrices with perturbed corners. By matrix decomposition with the Sherman–Morrison–Woodbury formula and constructing the corresponding displacement of matrices we derive the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type I in the form of products of Fermat numbers and some initial values. Furthermore, the properties of type II matrix can be also obtained, which benefits from the relation between type I and II matrices. Finally, we propose two algorithms for computing these properties and make some analysis about them to illustrate our theoretical results.

A new bound for the distance to singularity of a regular matrix pencil

AbstractFor regular matrix pencils 𝒜(s) = sE − A the distance to the nearest singular pencil in the Frobenius norm of the coefficients is called the distance to singularity. We derive a new lower bound for this distance by using the spectral theory of tridiagonal Toeplitz matrices. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On tropical eigenvalues of tridiagonal Toeplitz matrices

In this paper we determine the tropical algebraic eigenvalues of Toeplitz matrices. It is proved that these matrices have at most two distinct tropical eigenvalues and the explicit formulas for those eigenvalues are given. Also, the asymptotic behavior of eigenvalues of parameterized tridiagonal Toeplitz matrices over the classical field of complex numbers is shown.

The envelope of tridiagonal Toeplitz matrices and block-shift matrices

The envelope of a square complex matrix is a spectrum encompassing region in the complex plane. It is contained in and is akin to the numerical range in the sense that the envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curves, rather than half-planes. In this article, the geometry and properties of the envelopes of special matrices are examined. In particular, symmetries of the envelope of a tridiagonal Toeplitz matrix are obtained, and the envelopes of block-shift matrices, Jordan blocks and 2×2 matrices are explicitly characterized.

$$\mathcal O(n)$$ O ( n ) working precision inverses for symmetric tridiagonal Toeplitz matrices with $$\mathcal O(1)$$ O ( 1 ) floating point calculations

A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz matrices, i.e., matrices whose entries equal a on the diagonal and b on the extra diagonals (\(a, b\in \mathbb R\)). The inverses of such matrices are dense and there exist well known explicit formulas by which they can be calculated in \(\mathcal O(n^2)\). In this note we present a simplification of the problem that has proven to be rather useful in everyday practice: If \(\vert a\vert > 2\vert b\vert \), that is, if the matrix is strictly diagonally dominant, its inverse is a band matrix to working precision and the bandwidth is independent of n for sufficiently large n. Employing this observation, we construct a linear time algorithm for an explicit tridiagonal inversion that only uses \(\mathcal O(1)\) floating point operations. On the basis of this simplified inversion algorithm we outline the cornerstones for an efficient parallelizable approximative equation solver.

Modal Analysis of Weakly Coupled Electromagnetic Systems Using Tridiagonal Toeplitz Matrices

In the analysis of weakly coupled electromagnetic systems, consisting of several identical equidistant components, the coupling matrix is often approximated by tridiagonal Toeplitz matrices, whose eigenvalues and eigenvectors can be determined in closed-form. The theory of tridiagonal Toeplitz matrices is not new, it can be found in several linear algebra articles and math books, but its presentation is usually very dense and far from appealing to the electrical engineer. This paper has two goals. The first is of tutorial nature —we present a concise review of the modal analysis of tridiagonal matrices. The second goal is to call the reader’s attention to the inherent fragilities of the tridiagonal matrix approximation —we discuss and work out application examples, dealing with multiconductor transmission line systems, where the approximation is shown to clearly fail.

An explicit formula for the inverse of a pentadiagonal Toeplitz matrix

In this paper, we mainly consider finding an explicit formula for the inverse of a pentadiagonal Toeplitz matrix. For that purpose, we first factorize the modified form of a pentadiagonal Toeplitz matrix by two tridiagonal Toeplitz matrices, and then use the Sherman–Morrison–Woodbury inversion formula. As a result, an explicit inverse of a pentadiagonal Toeplitz matrix is obtained under certain assumptions. And numerical experiments are given to show the effectiveness of our results.

On the Gau–Wu number for some classes of matrices

For a given n×n matrix A, let k(A) stand for the maximal number of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range W(A). This number was recently introduced by Gau and Wu and we therefore call it the Gau–Wu number of the matrix A. We compute k(A) for two classes of n×n matrices A. A simple and explicit expression for k(A) for tridiagonal Toeplitz matrices A is derived. Furthermore, we prove that k(A)=2 for every pure almost normal matrix A. Note that for every matrix A we have k(A)⩾2, and for normal matrices A we have k(A)=n, so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the Gau–Wu number. Finally, matrices with maximal Gau–Wu number (k(A)=n) are considered.

Tridiagonal Toeplitz matrices: properties and novel applications

SUMMARYThe eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. This property is in the first part of the paper used to investigate the sensitivity of the spectrum. Explicit expressions for the structured distance to the closest normal matrix, the departure from normality, and theϵ‐pseudospectrum are derived. The second part of the paper discusses applications of the theory to inverse eigenvalue problems, the construction of Chebyshev polynomial‐based Krylov subspace bases, and Tikhonov regularization. Copyright © 2012 John Wiley & Sons, Ltd.

Analytical solutions for the resonance response of Goupillaud-type elastic media using [formula omitted]-transform methods

The resonance frequency spectrum is first derived for an m-layered Goupillaud-type elastic medium that is subjected to a discrete sinusoidal forcing function that varies harmonically with time at one end and held fixed at the other end. Analytical stress solutions are obtained from a coupled first-order system of difference equations using z-transform methods, where the determinant of the resulting global system matrix in the z-space |Am| is a palindromic polynomial with real coefficients. The zeros of the palindromic polynomial are distinct and are proven to lie on the unit circle for 1≤m≤5 and for certain classes of multilayered designs identified by tridiagonal Toeplitz matrices. An important result is the physical interpretation that all the positive angles, coterminal with the angles corresponding to the zeros of |Am| on the unit circle, represent the resonance frequency spectrum for the discrete model. A sequence of resonance frequencies for the discrete model appears to be universal for all multilayered designs with an odd number of layers, as it is independent of any design parameters.The resonance frequency results for the discrete model are then extended to describe the resonance frequency spectrum for the continuous model, where the forcing function applied at one end of the strip is continuous and varies harmonically with time while the other end is held fixed. The proposed natural frequency spectrum for a free-fixed m-layered Goupillaud-type strip is confirmed by independently solving a simplified form of the frequency equation, obtained after applying a transformation of the spatial variable. Our results suggest that the natural frequency spectrum depends on the layer impedance ratios and it is inversely proportional to the equal wave travel time for each layer.The results are used to identify layered designs with a common frequency spectrum and modify an existing design to obtain a desired frequency spectrum. Other connections are made with previous stress optimization results, the Chebyshev polynomials of the first and second kind, as well as the natural frequencies of a free-fixed non-Goupillaud-type layered strip.

Spectrum of certain tridiagonal matrices when their dimension goes to infinity

This paper deals with the spectra of matrices similar to infinite tridiagonal Toeplitz matrices with perturbations and with positive off-diagonal elements. We will discuss the asymptotic behavior of the spectrum of such matrices and we use them to determine the values of a matrix function, for an entire function. In particular we determine the matrix powers and matrix exponentials.