Positive Definite Toeplitz Completions

Bounds for the condition number are obtained which indicate the dependence of the conditioning of a symmetric positive definite Toeplitz matrix on the sign distribution of its reflection coefficients. This dependence is illustrated with numerical examples. Similar bounds for symmetric positive definite block Toeplitz matrices are also presented.

This paper is concerned with the determination of a close real banded positive definite Toeplitz matrix in the Frobenius norm to a given square real banded matrix. While it is straightforward to determine the closest banded Toeplitz matrix to a given square matrix, the additional requirement of positive definiteness makes the problem difficult. We review available theoretical results and provide a simple approach to determine a banded positive definite Toeplitz matrix.

A Schur-based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix

Positive definite completions of partial Hermitian matrices

Various preconditioners for symmetric positive-definite (SPD) Toeplitz matrices in circulant matrix form have recently been proposed. The spectral properties of the preconditioned SPD Toeplitz matrices have also been studied. In this research, Strang's preconditioner SN and our preconditioner KN are applied to an N N nonsymmetric (or nonhermitian) Toeplitz system TNX b. For a large class of Toeplitz matrices, it is proved that the singular values of SITN and KvlTN are clustered around unity except for a fixed number independent ofN. IfTN is additionally generated by a rational function, the eigenvalues of SITN and KITN can be characterized directly. Let the eigenvalues ofSITN andKITN be classified into the outliers and the clustered eigenvalues dependingon whether they converge to I asymptotically. Then, the number ofoutliers depends on the order of the rational generating function, and the clustering radius is proportional to the magnitude of the last elements in the generating sequence used to construct the preconditioner. Numerical experiments are provided to illustrate our theoretical study.

Algorithms for multiplication and inversion of ToepIitz matrices are presented that take advantage of the special structure of ToepIitz forms and the parallelism offered by concurrent processors. Multiplication of two general n×n Toeplitz matrices is defined on an array of 2n-1 processing elements. A block size may be chosen to effect an appropriate compromise between the number of processors and the amount of overhead due to interprocessor data communication. The parallel algorithm For inversion is based on a method of Justice For inverting a positive definite Toeplitz matrix. The computation, distributed over an array of n processing elements, generates a sequence of orthogonal polynomials from which the elements of the Toeplitz inverse are computed with O(n) concurrency.

Wideband spectrum sensing is a popular topic in signal processing, especially for many radar and communication applications. What we face is a high sampling rate and a large volume of samples, in which demand of reducing the sampling rate without sacrificing the sensing resolution and quality. The generalized coprime sampling can break the limitation of the Nyquist sampling theorem with both characteristics of sparse sensing and coprime numbers. To fully utilize all the information received of the derived correlation matrix constructed by the different time delays, the matrix completion method is exploited. The theory of matrix completion is an extension of compressive sensing, though, which is not restrained by the sparsity and the restricted isometry property. The interpolation-based method presented via the convex framework of the nuclear norm minimization has no extra fine-tuned parameters, which different from techniques like compressive covariance sampling, positive definite Toeplitz matrix completion, and so on. Moreover, compared to the selection-based method under a continuous set, the proposed method improves the spectral resolution and estimation accuracy to avoid the information losing. The Simulation results indicate the performance of the algorithm.

Sparse matrix decompositions and graph characterizations

A derivation of the Levinson algorithm for solving linear systems with symmetric positive definite Toeplitz matrix

Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property

The real positive definite completion problem for a 4-cycle

Cayley transform for Toeplitz and dual matrices

Fast O(n 2 ) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermonde-like matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast O(n 2 ) Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz ( Toeplitz-block ) matrices. The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires O(n 3 ) operations.

Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices

Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.