The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

Inverse properties of a class of seven-diagonal (near) Toeplitz matrices

This paper presents the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence explicit inverse formula is derived using the Sherman-Morrison formula. Related to the fixed-point iteration used to solve the differential equation, we show the positivity of the inverse matrix and construct an upper bound for the norms of the inverse matrix, which can be used to predict the convergence of the method.

The Explicit Formulae of the Determinants of Specific Pentadiagonal Toeplitz Hessenberg Matrices

The pentadiagonal Toeplitz matrix is a special kind of sparse matrix widely used in linear algebra, combinatorics, computational mathematics, and has been attracted much attention. We use the determinants of two specific Hessenberg matrices to represent the recurrence relations to prove two explicit formulae to evaluate the determinants of specific pentadiagonal Toeplitz matrices proposed in a recent paper [3]. Further, four new results are established.

Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation

We prove the wellposedness of a distributed-order space-fractional diffusion equation with variably distribution and its support, which could adequately model the challenging phenomena such as the anomalous diffusion in multiscale heterogeneous porous media, and smoothing properties of its solutions. We develop and analyze a collocation scheme for the proposed model based on the proved smoothing properties of the solutions. Furthermore, we approximately expand the stiffness matrix by a sum of Toeplitz matrices multiplied by diagonal matrices, which can be employed to develop the fast solver for the approximated system. We prove that it suffices to apply O(log N) terms of expansion to retain the accuracy of the numerical discretization of degree N, which reduces the storage of the stiffness matrix from O(N 2) to O(N log N), and the computational cost of matrix-vector multiplication from O(N 2) to O(N log2 N). Numerical results are presented to verify the effectiveness and the efficiency of the fast method.

A Novel Optimization Method for Bipolar Chaotic Toeplitz Measurement Matrix in Compressed Sensing

In this paper, a bipolar chaotic Toeplitz measurement matrix optimization algorithm for alternating optimization is presented. The construction of measurement matrices is one of the key techniques for compressive sensing from theory to engineering applications. Recent studies have shown that bipolar chaotic Toeplitz matrices, constructed by combining the intrinsic determinism of bipolar chaotic sequences with the advantages of Toeplitz matrices, have significant advantages over other measurement matrices in terms of memory overhead, computational complexity, and hard implementation. However, problems such as strong correlation and large interdependence coefficients between measurement matrices and sparse dictionaries may still exist in practical applications. To address this problem, we propose a new bipolar chaotic Toeplitz measurement matrix alternating optimization algorithm. Firstly, by introducing the structure matrix, the optimization problem of the measurement matrix is transformed into the optimization problem of the generating sequence, thus ensuring that the optimization process does not destroy the structural properties of the matrix; then, constraints are added to the values of the generating sequence during the optimization process, so that the optimized measurement matrix still maintains the bipolar properties. Finally, the effectiveness of the optimization algorithm in this paper is verified by simulation experiments. The experimental results show that the optimized bipolar chaotic Toeplitz measurement matrix can effectively reduce the reconstruction error and improve the reconstruction probability.