Hermitian Toeplitz Matrices Research Articles

Normal Toeplitz Matrices

It is well known from the work of Brown and Halmos [J. Reine Angew. Math., 213 (1963/1964), pp. 89–102] that an infinite Toeplitz matrix is normal if and only if it is a rotation and translation of a Hermitian Toeplitz matrix. In the present article we prove that all finite normal Toeplitz matrices are either generalised circulants or are obtained from Hermitian Toeplitz matrices by rotation and translation.

Localization of the Eigenvalues of Toeplitz Matrices using Additive Decomposition, Embedding in Circulants, and The Fourier Transform

This paper explores the relationship between Toeplitz and circulant matrices. Upper and lower bounds for all eigenvalues of hermitian Toeplitz matrices are given, capitalizing on the possibility of embedding a Toeplitz matrix in a circulant, and of expressing any n × n Toeplitz matrix as a sum of two matrices with known eigenvalues. The bounds can be simultaneously found using a single discrete Fourier transform evaluation. Thus, the total computation time is O (log2n) per bound. Simulation results indicate that the bounds are sharper than a few other known bounds.

Some Spectral Properties of Hermitian Toeplitz Matrices

Necessary conditions are given for the Hermitian Toeplitz matrix $T_n = ( t_{r - s} )_{r,s = 1}^n $ to have a repeated eigenvalue $\lambda $ with multiplicity $m > 1$ and for an eigenpolynomial of $T_n $ associated with $\lambda $ to have a given number of zeros off the unit circle $| z | = 1$. It is assumed that $t_r = \frac{1} {{2\pi }}\int_{ - \pi }^\pi {f( \theta )} e^{ - ir\theta } d\theta\, ( 0 \leq r \leq n - 1)$, where f is real-valued and in $L( - \pi ,\pi )$. The conditions are given in terms of the number of changes in sign of $f( \theta ) - \lambda $.

On inversion of hermitian block toeplitz matrices

A formula for the inverse of a hermitian block Toeplitz matrix via solutions of two block equations(instead of four in the general case) is given.

Gohberg-Semencul type formulas via embedding of Lyapunov equations (signal processing)

The authors present a new way of deriving Gohberg-Semencul-type inversion formulas for Hermitian Toeplitz and quasi-Toeplitz matrices. The approach is based on a certain Sigma -lossless embedding of Lyapunov equations. It has been shown that if a nonsingular matrix R has Toeplitz displacement inertia (p, q), then R/sup -1/ does not have the same Toeplitz displacement inertia. However, a para-Hermitian conjugate of R/sup -1/ will have this property. It is also shown that the Gohberg-Semencul-type inversion formulas can be formed directly in terms of certain parameters of the embedding. >

Some Aspects of Circulant Preconditioners

This paper studies circulant approximations of Hermitian Toeplitz matrices that are solutions of certain minimization problems relative to the Frobenius norm, the $lub_1 $, or the $lub_\infty $ norms. The examinations supplement the family of the so-called optimal and superoptimal preconditioners that have been proposed for the preconditioned conjugate gradient method for solving Toeplitz systems $T_n x = b$. For the new Frobenius norm approximations it is shown that they can be computed in $O(n\log (n))$ arithmetic operations, and that the eigenvalues of the preconditioned linear equations are asymptotically clustered around 1 if $T_n $ is a leading principal submatrix of an infinite Toeplitz matrix connected with a positive function of the Wiener class.

Recursive and iterative algorithms for computing eigenvalues of Hermitian Toeplitz matrices

The numerical solution of the complete eigenspectrum for Hermitian Toeplitz matrices is presented. Trench's algorithm (1989), which employs bisection on contiguous intervals, and the Pegasus method are used to achieve estimates of distinct eigenvalues. Several modifications of Trench's algorithm are examined; the goals are an increase in the rate of convergence, even at some reduction in estimate accuracy, and an accommodation of eigenvalue multiplicity or clustering. A promising approach that contains three key ingredients is found. They are: a modification of Trench's procedure to employ noncontiguous intervals, a procedure for multiplicity identification, and a replacement of the Pegasus method by the modified Rayleigh quotient iteration. The result is the basis for a novel eigenspectrum solver with a cubic convergence rate and good estimation accuracy. Simulation results for high-order Hermitian Toeplitz matrices are provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A Note on Computing Eigenvalues of Banded Hermitian Toeplitz Matrices

It is pointed out that the author’s $O(n^2 )$ algorithm for computing individual eigenvalues of an arbitrary $n \times n$ Hermitian Toeplitz matrix $T_n $ reduces to an $O(rn)$ algorithm if $T_n $ is banded, with bandwidth r.

On the extreme eigenvalues of Toeplitz and real Hankel interval matrices

The purpose of this paper is to evaluate the extreme eigenvalues of a Hermitian Toeplitz interval matrix and a real Hankel interval matrix. A (n×n)-dimensional Hermitian Toeplitz (HT) matrix is determined by the elements of its first row, sayr. If the elements ofr lie in complex intervals (i.e., rectangles of the complex plane), we call the resulting set of matrices an HT interval (HTI) matrix. An HTI matrix can model real world HT matrices where the elements of the vectorr have finite precision (e.g., because of quantization, or imprecise measurement devices). In this paper we prove that the extreme eigenvalues of a given HTI matrix can be easily obtained from the 22(n−1) vertex HT matrices where the first element ofr is set to zero. Similarly, as a special case we obtain that the extreme eigenvalues of a real symmetric Toeplitz interval (RSTI) matrix can be obtained from 2n−1 vertex matrices. Based on the above results we provide boxlike bounds for the eigenvalues on non-Hermitian complex and real Toeplitz interval matrices. Finally, we consider a real Hankel interval matrix. We prove that the maximal eigenvalue of a (n×n)-dimensional real Hankel interval matrix can be obtained from a subset of the vertex Hankel matrices containing 22n−3 vertex matrices, whereas the minimal eigenvalue can be obtained from another such subset also containing 22n−3 vertex matrices.

The RISE algorithm for recursive eigenspace decomposition

A recursive method for generating the eigenvalues and eigenvectors of Hermitian matrices is presented. This algorithm is closely related to recursive/iterative Toeplitz eigenspace decomposition (RITE) and thus may appropriately be named recursive/iterative self-adjoint eigenspace decomposition (RISE), since it is formulated for the more general Hermitian matrix. RISE may be performed at a cost of O(n/sup 3/) multiplies per matrix order, while RITE requires O(4n/sup 3/). It is demonstrated that RISE (as is also true with RITE) is numerically stable, and thus does not appreciably accumulate errors as the recursion progresses. Additionally, a variation of RITE is presented which allows the computation of the eigenvalues of successively smaller Hermitian Toeplitz matrices. The initial presentation of RITE only provided for the determination of the eigenvalues of successively larger matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Construction of a Hermitian Toeplitz matrix from an arbitrary set of eigenvalues

A solution to the inverse eigenvalue problem for Hermitian Toeplitz matrices is presented. The approach taken is to first construct a real symmetric negacyclic matrix of order 2n and to then relate the negacyclic matrix to a Hermitian Toeplitz matrix of order n with the desired eigenspectrum. >

Simple bounds on the extreme eigenvalues of Toeplitz matrices

Simple bounds are presented on the extreme eigenvalues of n*n-dimensional Hermitian Toeplitz matrices. Such a matrix, say T/sub n/, is determined by its first row. The proposed bounds have low complexity O(n); furthermore, examples are presented for which the proposed bounds are tighter than the Slepian-Landau bounds at their best, i.e. when the extreme eigenvalues of the submatrix obtained by deleting the first row and first column of T/sub n/ are known exactly. The bounds are first presented on the extreme eigenvalues of Hermitian Toeplitz matrices: the corresponding bounds for real symmetric Toeplitz matrices follow as a special case. Then, these bounds are extended to Hermitian Toeplitz interval matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new spectrum extension method that maximizes the multistep minimum prediction error-generalization of the maximum entropy concept

Given (n+1) consecutive autocorrelations of a stationary discrete-time stochastic process, how this finite sequence is extended so that the power spectral density associated with the resulting infinite sequence of correlations is nonnegative everywhere is discussed. It is well known that when the Hermitian Toeplitz matrix generated from the given autocorrelations is positive definite, the problem has an infinite number of solutions and the particular solution that maximizes the entropy functional results in a stable all-pole model of order n. Since maximization of the entropy functional is equivalent to maximization of the minimum mean-square error associated with one-step predictors, the problem of obtaining admissible extensions that maximize the minimum mean-square error associated with k-step (k<or=n) predictors, that are compatible with the given autocorrelations, is studied. It is shown that the resulting spectrum corresponds to that of a stable autoregressive moving average (ARMA) (n, k-1) process.<<ETX>>

Numerical solution of the eigenvalue problem for efficiently structured Hermitian matrices

An iterative procedure is proposed for computing the eigenvalues and eigenvectors of a class of specially structured Hermitian Toeplitz matrices which includes Hermitian Toeplitz and Toeplitz-plus-Hankel matrices. The computational cost per eigenvalue-eigenvector for a matrix of order n is O( n 2) in serial mode. Results of numerical experiments on Kac-Murdock-Szegö matrices and randomly generated real symmetric Toeplitz matrices and Toeplitz-plus-Hankel matrices of orders as high as 2000 are included.

An observation on certain spectral properties of Toeplitz matrices

Consider the generalized eigevalue problemAx=λBx,x=(xi), where A, B are Hermitian Toeplitz matrices. We give lower bounds to the number of the zeros of the «eigenpolynomial» U(z)=Σxizi, lying on the unit circle in the complex plane.

A Hermitian Toeplitz matrix is unitarily similar to a real Toeplitz-plus-Hankel matrix

Two unitary matrices are presented that transform a Hermitian Toeplitz matrix into a real Toeplitz-plus-Hankel matrix and vice versa. Additional properties and consequences of these unitary transformations are also presented. For a certain class of n-dimensional Toeplitz-plus-Hankel systems of equations, an efficient method of solution is presented. An example is given to illustrate the similarity transforms. Toeplitz-plus-Hankel systems arise in a wide variety of applications, such as linear filtering theory, discrete inverse scattering, and discretization of certain integral equations arising in mathematical physics.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Schur algorithms for Hermitian Toeplitz, and Hankel matrices with singular leading principal submatrices

It is shown how a simple matrix algebra procedure can be used to induce Schur-type algorithms for the solution of certain Toeplitz and Hankel linear systems of equations when given Levinson-Durbin algorithms for such problems. The algorithm of P. Delsarte et al. (1985) for Hermitian Toeplitz matrices in the singular case is used to induce a Schur algorithm for such matrices. An algorithm due to G. Heinig and K. Rost (1984) for Hankel matrices in the singular case is used to induce a Schur algorithm for such matrices. The Berlekamp-Massey algorithm is viewed as a kind of Levinson-Durbin algorithm and so is used to induce a Schur algorithm for the minimal partial realization problem. The Schur algorithm for Hermitian Toeplitz matrices in the singular case is shown to be amenable to implementation on a linearly connected parallel processor array of the sort considered by Kung and Hu (1983), and in fact generalizes their result to the singular case.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Structured invariant spaces of vector valued rational functions, hermitian matrices, and a generalization of the lohvidov laws

Finite dimensional indefinite inner product spaces of vector valued rational functions which are (1) invariant under the generalized backward shift and (2) subject to a structural identity, and subspaces and “superspaces” thereof are studied. The theory of these spaces is then applied to deduce a generalization of a pair of rules due to lohvidov for evaluating the inertia of certain subblocks of Hermitian Toeplitz and Hermitian Hankel matrices. The connecting link rests on the identification of a Hermitian matrix as the Gram matrix of a space of vector valued functions of the type considered in the first part of the paper. Corresponding generalizations of another pair of theorems by lohvidov on the rank of certain subblocks of non-Hermitian Teoplitz and non-Hermitian Hankel matrices are also stated, but the proofs will be presented elsewhere.

Computationally Efficient Reduced Polynomial Based Algorithms for Hermitian Toeplitz Matrices

In this work, the mathematical structure is analyzed that is associated with the fast order-recursive algorithms for computing the reflection coefficients, and the Levinson polynomial associated with a hermitian, positive-definite Toeplitz matrix. A new form of three-term recurrence relation is derived anda computationally efficient alternative to the classical Levinson–Durbin algorithm is obtained. The computational complexity of the new algorithm is the same as those of the split algorithms described in the recent literature. The new algorithm also provides further insight into the mathematical properties of the structurally rich Toeplitz matrices.

OnH-contractions and the extension problem for hermitian block toeplitz matrices

It is well known that any positive semi-definite Hermitian block Toeplitz matrix Tn (C0 ,C1 ,…Cn ) has positive semi-definite Toeplitz extensions Tn+1 (C0 ,…Cn Cn+1 ). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices Tn to matrices Tn+1 with the same number of negative eigenvalues as Tn . A necessary and sufficient condition for the existence of such Cn+1 is derived, and a parametrization of all corresponding Cn+1 is presented. Our approach to the Hermitian block Toeplitz extension problem is based on a formulation as a completion problem for contractions in indefinite inner product spaces. A complete solution of this more general problem is given.