An inverse problem for Toeplitz matrices

A simplified solution to an inverse problem for Toeplitz matrices using central mass sequences is presented. Some connections with discrete transmission lines are mentioned.     

Eigenvectors of certain matrices

An algorithm is given for calculation of eigenvalues and eigenvectors of centrosymmetric and some related matrices, and some desirable properties of the algorithm are proved. Centrosymmetric matrices are characterized by a symmetry property of their eigenvectors and this result is used to establish a property of certain methods for the numerical solution of differential equations.     

On the eigenvalue problem for Toeplitz band matrices

A formula is given for the characteristic polynomial of an nth order Toeplitz band matrix, with bandwidth k < n, in terms of the zeros of a kth degree polynomial with coefficients independent of n. The complexity of the formula depends on the bandwidth k, and not on the order n. Also given is a formula for eigenvectors, in terms of the same zeros and k coefficients which can be obtained by solving a k × k homogeneous system.     

On inverses of Hessenberg matrices

The lower half of the inverse of a lower Hessenberg matrix is shown to have a simple structure. The result is applied to find an algorithm for finding the inverse of a tridiagonal matrix. With minor modifications, the technique applies to block Hessenberg matrices.     

A class of test Hessenberg matrices

We present a class of parametric Hessenberg matrices, intended for testing linear-algebraic procedures. Their eigenvalues and subdiagonal elements are arbitrarily prescribed, while the eigenvector and the inverse matrices are computed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 89–92, 1981.     

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

We consider banded block Toeplitz matrices T_n with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of T_n for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of T_n and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if T_n is a Hessenberg matrix.     

On the trace approximation problem for truncated Toeplitz operators and matrices

The paper is devoted to the problem of approximation of the traces of products of truncated Toeplitz operators and matrices generated by integrable real symmetric functions defined on the real line (resp. on the unit circle), and estimation of the corresponding errors. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous- and discrete-time stationary processes (asymptotic distributions and large deviations of Toeplitz type quadratic functionals and forms, parametric and nonparametric estimation, etc.)We review and summarize the known results concerning the trace approximation problem and prove some new results.     

Polynomial perturbations of bilinear functionals and Hessenberg matrices

This paper deals with symmetric and non-symmetric polynomial perturbations of symmetric quasi-definite bilinear functionals. We establish a relation between the Hessenberg matrices associated with the initial and the perturbed functionals using LU and QR factorizations. Moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals.     

Computing powers of arbitrary Hessenberg matrices

An efficient algorithm for the computation of powers of an n × n arbitrary lower Hessenberg matrix is presented. Numerical examples are used to show the computational details. A comparison of the algorithm with two other methods of matrix multiplication proposed by Brent and by Winograd is included. Related algorithms were proposed earlier by Datta and Datta for lower Hessenberg matrices with unit super-diagonal elements, and by Barnett for the companion matrix.     

The problem of the extensions of a positve definite sequence of second-order Toeplitz matrices

Some questions regarding the extension of positive definite sequences of Toeplitz matrices are considered. By the methods of the j-theory the possibility of the extension of such sequences is established for second-order matrices and the structure of a stepwise extension is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1283–1286, September, 1990.     

Hessenberg matrices and the Pell and Perrin numbers

In this paper, we investigate the Pell sequence and the Perrin sequence and we derive some relationships between these sequences and permanents and determinants of one type of Hessenberg matrices.     

An inverse problem for Toeplitz matrices and the synthesis of discrete transmission lines

Let A_m be a positive definite, m x m, Toeplitz matrix. Let A_k be its k x k principal minor (for any k?m), which is also positive definite and Toeplitz. Define the central mass sequence {?₁,…,?_m} by ?_k = sup{?: A_k ? ?Π_k > 0}, in which Π_k is the k x k matrix of all 1's. We show how knowledge of the sequence {?_k} is equivalent to knowledge of the matrix A_m. This result has application to the direct and inverse problems for a transmission line which consists of piecewise constant components. Knowing the impulse response of the transmission line, we can calculate the capacitance taper of the line, and vice versa.     

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Let a, b and c be fixed complex numbers. Let M _n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M _n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M _n (a, b, c). We also show that all complex polynomials in M _n (a, b, c) are Toeplitz matrices. In particular, the inverse of M _n (a, b, c) is a Toeplitz matrix when it exists.