Lower Triangular Toeplitz Matrices Research Articles

Polynomial Wiener LQG Controllers based on Toeplitz Matrices

This paper re-examines the discrete-time Linear Quadratic Gaussian (LQG) regulator problem. The normal approach to this problem is to use a Kalman filter state estimator and Kalman control state feedback. Though quite successful, an alternative approach in the frequency domain was employed later. That method used z-transfer functions or polynomials in the z-domain. The transfer function approach is similar to the method used in Wiener filtering and requires the use of Diophantine equations (sometimes bilateral) to find the optimal controller. The contribution here uses a similar approach but uses lower triangular Toeplitz matrices instead of polynomials to gain advantage of eliminating the use of Diophantine equations. This is because the single Diophantine equation approach fails when the system has non-relative prime polynomials and the need for bilateral Diophantine equations is computationally far more complex.

On the solutions of the equation AXB = C under Toeplitz‐like and Hankel matrices constraint

In this paper, we are mainly concerned with 2 types of constrained matrix equation problems of the form AXB=C, the least squares problem and the optimal approximation problem, and we consider several constraint matrices, such as general Toeplitz matrices, upper triangular Toeplitz matrices, lower triangular Toeplitz matrices, symmetric Toeplitz matrices, and Hankel matrices. In the first problem, owing to the special structure of the constraint matrix , we construct special algorithms; necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions. In the second problem, we use von Neumann alternating projection algorithm to obtain the solutions of problem. Then we give 2 numerical examples to demonstrate the effectiveness of the algorithms.

Schur Parameters, Toeplitz Matrices, and Kreĭn Shorted Operators

We establish connections between Schur parameters of the Schur class operator-valued functions, the corresponding simple conservative realizations, lower triangular Toeplitz matrices, and Kreĭn shorted operators. By means of Schur parameters or shorted operators for defect operators of Toeplitz matrices necessary and sufficient conditions for a simple conservative discrete-time system to be controllable/observable and for a completely non-unitary contraction to be completely non-isometric/completely non-co-isometric are obtained. For the Schur problem a characterization of central solution and uniqueness criteria to the solution are given in terms of shorted operators for defect operators of contractive Toeplitz matrices, corresponding to data.

An optimal bound for inverses of triangular matrices with monotone entries

This article provides a new bound for 1-norms of inverses of positive triangular matrices with monotonic column entries. The main theorem refines a recent inequality established in Vecchio and Mallik [Bounds on the inverses of non-negative lower triangular Toeplitz matrices with monotonicity properties, Linear Multilinear Alg., 55 (2007), pp. 365–379]. The results are shown to be in a sense best possible under the given constraints.

Inverting linear combinations of identity and generalized Catalan matrices

We introduce the notion of the generalized Catalan matrix as a kind of lower triangular Toeplitz matrix whose nonzero elements involve the generalized Catalan numbers. Inverse of the linear combination of the Pascal matrix with the identity matrix is computed in Aggarwala and Lamoureux (2002) [1]. In this paper, continuing this idea, we invert various linear combinations of the generalized Catalan matrix with the identity matrix. A simple and efficient approach to invert the Pascal matrix plus one in terms of the Hadamard product of the Pascal matrix and appropriate lower triangular Toeplitz matrices is considered in Yang and Liu (2006) [14]. We derive representations for inverses of linear combinations of the generalized Catalan matrix and the identity matrix, in terms of the Hadamard product which includes the Generalized Catalan matrix and appropriate lower triangular Toeplitz matrix.

A Bound for the Inverse of a Lower Triangular Toeplitz Matrix

The expression of a bound of the uniform norm of infinite lower triangular Toeplitz matrices with nonnegative entries is found. All the results are obtained by studying the behavior of the resolvent kernel and of the fundamental matrix of the recurrence relation, which generates the sequence of the entries of the considered matrix.

Triangular Toeplitz contractions and Cowen sets for analytic polynomials

Let $\mathfrak {L}_{N}$ be the collection of $N\times N$ lower triangular Toeplitz matrices and let $\mathfrak {T}_{N}$ be the collection of $N\times N$ lower triangular Toeplitz contractions. We show that $\mathfrak {T}_{N}$ is compact and strictly convex, in the spectral norm, with respect to $\mathfrak {L}_{N}$; that is, $\mathfrak {T}_{N}$ is compact, convex and $\partial _{\mathfrak {L}_{N}} \mathfrak {T}_{N} \subseteq \operatorname {ext}\mathfrak {T}_{N}$, where $\partial _{\mathfrak {L}_{N}}(\cdot )$ and $\operatorname {ext}(\cdot )$ denote the topological boundary with respect to $\mathfrak {L}_{N}$ and the set of extreme points, respectively. As an application, we show that the reduced Cowen set for an analytic polynomial is strictly convex; more precisely, if $f$ is an analytic polynomial and if $G_f’ := \{\, g\in H^\infty (\mathbb {T}): g(0)=0$ and the Toeplitz operator $T_{f+\bar {sg}}$ is hyponormal$\,\}$, then $G_{f}’$ is strictly convex. This answers a question of C. Cowen for the case of analytic polynomials.

Computations with Gohberg-Semencul-type formulas for Toeplitz matrices

The inverse of a Toeplitz matrix T n can be represented in different ways by Gohberg-Semencul-type formulas as the sum of products of upper and lower triangular Toeplitz matrices. If we have given such a formula, we can solve every equation T nξ = b in O( n log n) steps. There are three main questions arising with such representations of T n −1: (1) which special linear equations can be solved in order to get generating vectors for Gohberg-Semencul-type formulas, (2) which algorithm we want to apply to solve these questions, and (3) which special Gohberg-Semencul-type formula we want to use for evaluating T n −1 b. In this paper we present an elementary approach to derive all Gohberg-Semencul-type formulas. Then we introduce representations of T n −1 with special properties. In particular we prove that there exists a Gohberg-Semencul-type formula such that the generating vectors are pairwise orthogonal. Finally, based on the previous results, we give new fast and stable algorithms for solving linear Toeplitz systems that can be used to compute generating vectors for Gohberg-Semencul-type formulas. In the case of a breakdown or near-breakdown in the kth step, the new method introduces a perturbation in the kth entry t k of T n such that the perturbed submatrix T̂ k is well conditioned. By using the Sherman-Morrison-Woodbury formula it is possible to recover the original problem as soon as the corresponding submatrix T k + r is again nonsingular.

Computations with Gohberg-Semencul-type formulas for Toeplitz matrices

The inverse of a Toeplitz matrix Tn can be represented in different ways by Gohberg-Semencul-type formulas as the sum of products of upper and lower triangular Toeplitz matrices. If we have given such a formula, we can solve every equation Tnξ = b in O(n log n) steps. There are three main questions arising with such representations of Tn−1: (1) which special linear equations can be solved in order to get generating vectors for Gohberg-Semencul-type formulas, (2) which algorithm we want to apply to solve these questions, and (3) which special Gohberg-Semencul-type formula we want to use for evaluating Tn−1b. In this paper we present an elementary approach to derive all Gohberg-Semencul-type formulas. Then we introduce representations of Tn−1 with special properties. In particular we prove that there exists a Gohberg-Semencul-type formula such that the generating vectors are pairwise orthogonal. Finally, based on the previous results, we give new fast and stable algorithms for solving linear Toeplitz systems that can be used to compute generating vectors for Gohberg-Semencul-type formulas. In the case of a breakdown or near-breakdown in the kth step, the new method introduces a perturbation in the kth entry tk of Tn such that the perturbed submatrix T̂k is well conditioned. By using the Sherman-Morrison-Woodbury formula it is possible to recover the original problem as soon as the corresponding submatrix Tk + r is again nonsingular.

Displacement operator based decompositions of matrices using circulants or other group matrices

We show how an arbitrary square matrix can be expressed as sums of products of circulant and upper or lower triangular Toeplitz matrices, and as sums of products of matrices derived from finite groups (group matrices) and matrices which are “close” to group matrices. The results obtained are interesting from several points of view: they lead to different methods for computing linear transforms using FFTs or fast convolution algorithms, to faster methods for solving Toeplitz systems, to potential methods for mapping linear transforms to parallel computer architectures with interconnection scheme given by the group graph of a finite group, and possibly to matrix theoretic methods for expressing relationship between finite groups.