Triangular Factorization of Structured Hermitian Matrices

Some identities for determinants of structured matrices

In this work, we consider the parametric data model employed in applications such as line spectral estimation and direction-of-arrival estimation. We focus on the stochastic maximum likelihood estimation (MLE) framework and offer approaches to estimate the parameter of interest in a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gridless</i> manner, overcoming the model complexities of the past. This progress is enabled by the modern trend of reparameterization of the objective, and exploiting the sparse Bayesian learning (SBL) approach. The latter is shown to be a correlation-aware method, and for the underlying problem it is identified as a grid-based technique for recovering a structured covariance matrix of the measurements. For the case when the structured matrix is expressible as a sampled Toeplitz matrix, such as when measurements are sampled in time or space at regular intervals, additional constraints and reparameterization of the SBL objective leads to the proposed structured matrix recovery technique based on MLE. The proposed optimization problem is non-convex, and we propose a majorization-minimization based iterative procedure to estimate the structured matrix; each iteration solves a semidefinite program. We recover the parameter of interest in a gridless manner by appealing to the Carathéodory-Féjer result on decomposition of positive semidefinite (PSD) Toeplitz matrices. For the general case of irregularly spaced time or spatial samples, we propose an iterative SBL procedure that refines grid points to increase resolution near potential source locations, while maintaining a low per iteration complexity. We provide numerical results to evaluate and compare the performance of the proposed techniques with other gridless techniques, and the Cramér-Rao bound. The proposed correlation-aware approach is more robust to environmental/system effects such as low number of snapshots, correlated sources, small separation between source locations and improves sources identifiability.

Many engineering or mathematical problems require to factorize structured matrices (Toeplitz, Hankel, Vandermonde, products of such matrices and their inverses, Schur complements, etc.) either in explicit or in disguised form. Consequently there exist various analytic tools regarding structured matrices as well as several factorization algorithms. In this thesis, we show that many of these results and several significant generalizations can be obtained in a very constructive way. The generic form is to use elementary circular and hyperbolic transformations to triangularize a certain array of numbers derived from the displacement representation of the given structured matrix; the desired results can then be read off from the resulting array. These fast array algorithms require $O$($mn$) operations for LU and QR factorizations of $m$ x $n$ structured matrices, and $O$($mn$) or even $O$($n$log$\sp2n$) operations for solving matrix equations. Also the array form suggests various alternative algorithms, depending upon the order in which the transformations are applied; these variations can have different numerical properties and lead to different implementations. Our algorithm is based on a generalized definition of displacement for block-Toeplitz (Hankel) and Toeplitz (Hankel)-block matrices slightly extending the previous definitions of Kailath, Kung and Morf (1979) and Lev-Ari and Kailath (1984). An important property of displacement structure is that it is preserved under Schur complementations. It will turn out that Toeplitz-(Hankel)-derived (near-Toeplitz, Toeplitz-like, etc.) matrices are perhaps best regarded as particular Schur complements obtained from suitably defined block matrices. The displacement structure is used to obtain a generalized Schur algorithm for the triangular and orthogonal factorizations of all such matrices, and well structured solutions of the corresponding exact and overdetermined systems of linear equations.

Permutability of Toeplitz and Hankel matrices

CONTENTS Introduction Chapter I. The algebraic structure of the spaces of Toeplitz and Hankel matrices § 1. Basic definitions § 2. Basic notation § 3. Statement of the theorems § 4. Auxiliary lemmas § 5. Proof of the theorems Chapter II. Fast computations in some problems of linear algebra connected with Toeplitz and Hankel matrices § 1. Basic definitions and notation § 2. Statement of the theorems § 3. Auxiliary lemmas § 4. Proof of the theorems Chapter III. The fast prediction of stochastic processes § 1. The fast prediction of stationary stochastic processes § 2. The fast prediction of piecewise constant random functions Chapter IV. The structure of Toeplitz and Hankel integral operators § 1. Definitions § 2. Notation § 3. Statement of the theorems References

On algebras of Toeplitz plus Hankel matrices

Conditions for commuting a Toeplitz matrix and a Hankel matrix were obtained relatively recently (in 2015). The solution to the problem of describing all anti-commuting pairs (T, H), where T is a Toeplitz matrix and H is a Hankel matrix, is sketched below.

This paper is a survey of some works on paracommutators and Hankel operators in China, which are the subjects I have been studying since 1985. The classical Hankel operator is defined on the Hilbert space l 2 by the Hankel matrix Γ = {b n+k } n,k ≥0, and the Toeplitz operator is defined also on l 2 by the toeplitz matrix T = {b n−k } n≥k≥0. By the theory of the Fourier series both of them become the operators on the Hardy space \( {{H}^{2}}(T) = \{ f \in {{L}^{2}}(T):\hat{f}(n) = 0,n < 0\} . \). Let \( \overline H ^2 (T) \) denote the conjugate Hardy space and P,P denote the ortnogonal projections from L 2(T) into H 2(T),H 2 (T) respectively. Then the Toeplitz operator is defined as follows $$ {T_b}(f) = P(\bar bf),for f \in {H^2}(T) $$ (1.1) and the Hankel operator is defined as follows $$ {H_b}(f) = \bar P(\bar bf),for f \in {H^2}(T).$$ (1.2)

Factorizations of Hermitian block Hankel matrices

Kernel structure of block Hankel and Toeplitz matrices and partial realization

We study non-interacting fermionic systems dissipatively driven at their boundaries, focusing in particular on the case of a non-number-conserving Hamiltonian, which for example describes an $XY$ spin chain. We show that despite the lack of number conservation, it is possible to convert the problem of calculating the normal modes of the master equations and their corresponding rapidities, into diagonalizing simply an $L\times L$ tridiagonal bordered $2-$Toeplitz matrix, where $L$ is the size of the system. Such structure of matrix allows us to further reduce the problem into solving a scalar trigonometric non-linear equation for which we also show, in the case of an Ising chain, exact analytical explicit, and system size independent, solutions.

Products of Hankel and Toeplitz Operators on the Bergman Space

Averaging operations are considered in connection with exponential splitting methods. Toeplitz plus Hankel related matrices are resplit by applying appropriate averaging operators leading to a hierarchy of structured matrices. With the resulting parts, the option of using exponential splitting methods becomes available. A related, seemingly important group of unitary unipotents is looked at. Based on a formula due to Lenard, a very fast iterative method to find the nearest Toeplitz plus Hankel matrix in the Frobenius norm is devised.

The paper presents fast algorithms for designing finite impulse response (FIR) notch filters. The aim is to design a digital FIR notch filter so that the magnitude of the filter has a deep notch at a specified frequency, and as the notch frequency changes, the filter coefficients should be able to track the notch fast in real time. The filter design problem is first converted into a convex optimization problem in the autocorrelation domain. The frequency response of the autocorrelation of the filter impulse response is compared with the desired filter response and the integral square error is minimized with respect to the unknown autocorrelation coefficients. Spectral factorization is used to calculate the coefficients of the filter. In the optimization process, the computational advantage is obtained by exploiting the structure of the Hessian matrix which consists of a Toeplitz plus a Hankel matrix. Two methods have been used for solving the Toeplitz‐plus‐Hankel system of equations. In the first method, the computational time is reduced by using Block–Levinson's recursion for solving the Toeplitz‐plus‐Hankel system of matrices. In the second method, the conjugate gradient method with different preconditioners is used to solve the system. Comparative studies demonstrate the computational advantages of the latter. Both these algorithms have been used to obtain the autocorrelation coefficients of notch filters with different orders. The original filter coefficients are found by spectral factorization and each of these filters have been tested for filtering synthetic as well as real‐life signals. Copyright © 2007 John Wiley &amp; Sons, Ltd.

We say that square matrices A and B of the same order quasi-commute if AB = σBA for a certain scalar σ. The classical definitions of commutation and anti-commutation are particular cases of the above definition. We give a complete description of pairs of quasi-commuting Toeplitz and Hankel matrices for σ ≠ ±1.