The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $ A_1XA_1^\ast = C_1 $ and $ A_2XA_2^\ast = C_2 $

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>

Two Explicit Characterizations of the General Nonnegative-Definite Covariance Matrix Structure for Equality of BLUEs, WLSEs, and LSEs

We provide a new, concise derivation of necessary and sufficient conditions for the explicit characterization of the general nonnegative-definite covariance structure V of a general Gauss-Markov model with E(y) and Var(y) such that the best linear unbiased estimator, the weighted least squares estimator, and the least squares estimator of X&beta; are identical. In addition, we derive a representation of the general nonnegative-definite covariance structure V defined above in terms of its Moore-Penrose pseudo-inverse.

Parity symmetry in a number of problems of quantum and structural chemistry

A synthetic review and new results are given of the alternant symmetry theory and its applications within a unified approach. It is based on J–symmetry (parity) operators. Unlike usual commutation rules, these symmetry operators anticommute with Hamiltonians or other relevant quantities. In the J–symmetry terms we treat a variety of problems and topics, mainly related to π-shells of conjugated molecules. In particular, various orbital theories are outlined with a systematic use of block-matrix technique (density matrices, operator functions etc.). Noval π‑models and their J–symmetry are studied within the current context of single-molecule conductance and the relevant problems concerning Green’s function and electron transmission evaluation. We stress on the key importance of account for π-electron correlation for describing correctly transmission π-spectra. We discuss electron-structure peculiarities of alternant radical states and the validity of the Lieb-Ovchinnikov spin rule resulting from the J–symmetry and electron correlation effects. It is shown how the simplified (based on Hückel’s MOs) spin-polarized theory provides a correct number of effectively unpaired electrons in polyradicaloid alternant molecules. Another type of problems is concerned with chirality (generllly, structural asymmetry) problems. By spectral analysys of the previously defined chirality operator we could reinterpret the problem in terms of J–symmetry. It allowed us to construct here the noval chirality operator which is nonnegative definite and vanishes on achiral structures. Its simplest invariant, the matrix trace, surves us as a quantitative measure of the structural (electronic) chirality. Preliminary calculations tell us that the new chirality index behaves reasonably even for the difficult (high-symmetry) chiral systems.

On Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices

We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.

A Noninterpolated Estimate of Horizontal Spatial Covariance from Nonorthogonally and Irregularly Sampled Scalar Velocities

This paper presents a least squares method to estimate the horizontal (isotropic or anisotropic) spatial covariance of two-dimensional orthogonal vector components, without introducing an intervening mapping step and biases, from the spatial covariance of the nonorthogonally and irregularly sampled raw scalar velocities. The field is assumed to be locally homogeneous in space and sampled in an ensemble so the unknown spatial covariance is a function of spatial lag only. The transformation between the irregular grid on which nonorthogonal scalar projections of the vector are sampled and the regular orthogonal grid on which they will be mapped is created using the geometry of the problem. The spatial covariance of the orthogonal velocity components of the field is parameterized by either the energy (power) spectrum in the wavenumber domain or the lagged covariance in the spatial domain. The energy spectrum is constrained to be nonnegative definite as part of the solution of the inverse problem. This approach is applied to three example sets of data, using nonorthogonally and irregularly sampled radial velocity data obtained from 1) a simple spectral model, 2) a regional numerical model, and 3) an array of high-frequency radars. In tests where the true covariance is known, the proposed direct approaches fitting to parameterizations of the nonorthogonally and irregularly sampled raw data in the wavenumber domain and spatial domain outperform methods that map the data to a regular grid before estimating the covariance.

An orthogonal power method of solving the partial eigenproblem for a symmetric nonnegative definite matrix

Предложена и обоснована экономичная версия метода сопряженных направлений для построения нетривиального решения однородной системы линейных алгебраических уравнений с вырожденной симметричной неотрицательно определенной квадратной матрицей. Предложено однопараметрическое семейство одношаговых нелинейных итерационных процессов вычисления собственного вектора, отвечающего наибольшему собственному значению симметричной неотрицательно определенной квадратной матрицы. Это семейство включает в себя степенной метод как частный случай. Доказана сходимость возникающих последовательностей векторов к собственному вектору, ассоциированному с наибольшим характеристическим числом матрицы. Предложена двухшаговая процедура ускорения сходимости итераций этих процессов, в основе которой лежит ортогонализация в подпространстве Крылова. Приведены результаты численных экспериментов. An efficient version of the conjugate direction method to find a nontrivial solution of a homogeneous system of linear algebraic equations with a singular symmetric nonnegative definite square matrix is proposed and substantiated. A one-parameter family of one-step nonlinear iterative processes to determine the eigenvector corresponding to the largest eigenvalue of a symmetric nonnegative definite square matrix is also proposed. This family includes the power method as a special case. The convergence of corresponding vector sequences to the eigenvector associated with the largest eigenvalue of the matrix is proved. A two-step procedure is formulated to accelerate the convergence of iterations for these processes. This procedure is based on the orthogonalization in Krylov subspaces. A number of numerial results are discussed.