Tridiagonal Toeplitz matrices: properties and novel applications

Abstract

The 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 the "-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.TRIDIAGONAL TOEPLITZ MATRICES 303 detail in [13][14][15]. Our interest in tridiagonal Toeplitz matrices stems from the possibility of deriving explicit formulas for quantities of interest and from the many applications of these matrices.This paper is organized as follows. The eigenvalue sensitivity is investigated in Sections 2-6. Numerical illustrations also are provided. The latter part of this paper describes a few applications that are believed to be new. We consider an inverse eigenvalue problem in Section 7, where we also introduce a minimization problem, whose solution is a trapezoidal tridiagonal Toeplitz matrix. The latter matrices can be applied as regularization matrices in Tikhonov regularization. This application is described in Section 8. Section 9 is concerned with the construction of nonorthogonal Krylov subspace bases based on the recursion formulas for suitably chosen translated and scaled Chebyshev polynomials. The use of such bases in Krylov subspace methods for the solution of large linear systems of equations or for the computation of a few eigenvalues of a large matrix is attractive in parallel computing environments that do not allow efficient execution of the Arnoldi process for generating an orthonormal basis; see [16][17][18] for discussions. We describe how tridiagonal Toeplitz matrices can be applied to determine a suitable interval on which the translated and scaled Chebyshev polynomials are required to be orthogonal. Concluding remarks can be found in Section 10.

Physical Layer Network Security in the Full-Duplex Relay System

“…where θ b is the bth eigenvalue of H H H. Because H is a Toeplitz matrix, θ b is given by (see [25])…”

Inverse problems for regularization matrices

“…The situation when k = 1 has previously been discussed in [15]. In such a case B is an upper trapezoidal tridiagonal Toeplitz matrix and, according to Proposition 2.1, the two columns of the matrix in (2.2) are linearly dependent if and only if the components of x satisfy ξ h = αξ h+2 , h = 1 : n − 2, for some α ∈ C. Hence, in the tridiagonal case one has necessary and sufficient conditions for the unicity of the solution of the least-squares problem in (2.2).…”

Fast Approximation for Toeplitz, Tridiagonal, Symmetric and Positive Definite Linear Systems that Grow Over Time

“…In many cases the size of these systems is very large; then, computing the solution in a reasonable amount of time becomes a problem for real time applications [9], [10], [11], [12], [13], [14], [3], [15]. Due to the increasing interest in tridiagonal matrices, it is important to understand its properties and its applications [16], [12], [17], [18], [19], [20], [21], [22], [23], [24].…”

“…A ij = 0 for |i − j| > 1, there exists fast algorithms to solve it in O(n) time [16], [11], [2], [19], [3], [25], [24]. Additionally, if A is Toeplitz, symmetric and positive definite as shown in (1), the computation can be optimized [16], [11], [2], [3], for example by using the Cholesky decomposition A = LL T [11], [3].…”

“…When the matrix A has the structure defined in (1), the coefficients can be computed in a closed form [16], [19], as shown in (4). Those coefficients have the following limits:…”