Showing papers on "Matrix (mathematics) published in 1996"

A review of target decomposition theorems in radar polarimetry

Abstract: In this paper, we provide a review of the different approaches used for target decomposition theory in radar polarimetry. We classify three main types of theorem; those based on the Mueller matrix and Stokes vector, those using an eigenvector analysis of the covariance or coherency matrix, and those employing coherent decomposition of the scattering matrix. We unify the formulation of these different approaches using transformation theory and an eigenvector analysis. We show how special forms of these decompositions apply for the important case of backscatter from terrain with generic symmetries.

Equivariant adaptive source separation

Abstract: Source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed. This paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence (EASI). The EASI algorithms are based on the idea of serial updating. This specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures. Most importantly, the performance of an EASI algorithm does not depend on the mixing matrix. In particular, convergence rates, stability conditions, and interference rejection levels depend only on the (normalized) distributions of the source signals. Closed-form expressions of these quantities are given via an asymptotic performance analysis. The theme of equivariance is stressed throughout the paper. The source separation problem has an underlying multiplicative structure. The parameter space forms a (matrix) multiplicative group. We explore the (favorable) consequences of this fact on implementation, performance, and optimization of EASI algorithms.

Interpretation of Mueller matrices based on polar decomposition

Abstract: We present an algorithm that decomposes a Mueller matrix into a sequence of three matrix factors: a diattenuator, followed by a retarder, then followed by a depolarizer. Those factors are unique except for singular Mueller matrices. Based on this decomposition, the diattenuation and the retardance of a Mueller matrix can be defined and computed. Thus this algorithm is useful for performing data reduction upon experimentally determined Mueller matrices.

Handbook of Matrices

Abstract: Definitions, Notations, Terminology. Rules for Matrix Operations. Matrix Valued Functions of a Matrix. Trace, Determinant and Rank of a Matrix. Eigenvalues and Singular Values. Matrix Decompositions and Canonical Forms. Vectorization Operators. Vector and Matrix Norms. Properties of Special Matrices. Vector and Matrix Derivatives. Polynomials, Power Series and Matrices. Appendix. References. Index.

Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings

Abstract: Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings. Many implementation variants of the algorithms are also presented. Their physical interpretations are given, and their numerical stabilities and efficiencies are discussed in detail. The single most important criterion for achieving unconditional numerical stability with both algorithms is to avoid the exponentially growing functions in every step of the matrix recursion. From the viewpoint of numerical efficiency, the S-matrix algorithm is generally preferred to the R-matrix algorithm, but exceptional cases are noted.

On convergence properties of the em algorithm for gaussian mixtures

Abstract: We build up the mathematical connection between the “Expectation-Maximization” (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix P, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of P and provide new results analyzing the effect that P has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of gaussian mixture models.

Time-dependent local-density approximation in real time

Abstract: We study the dipole response of atomic clusters by solving the equations of the time-dependent local-density approximation in real time. The method appears to be more efficient than matrix or Green's function methods for large clusters modeled with realistic ionic pseudopotentials. As applications of the method, we exhibit results for sodium and lithium clusters and for ${\mathrm{C}}_{60}$ molecules. The calculated Mie resonance in ${\mathrm{Na}}_{147}$ is practically identical to that obtained in the jellium approximation, leaving the origin of the redshift unresolved. The pseudopotential effects are strong in lithium and act to broaden the Mie resonance and give it a substantial redshift, confirming earlier studies. There is also a large broadening due to Landau damping in the calculated ${\mathrm{C}}_{60}$ response, again confirming earlier studies. \textcopyright{} 1996 The American Physical Society.

An Interior-Point Method for Semidefinite Programming

Abstract: We propose a new interior-point-based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices....

Closed-form 2-D angle estimation with rectangular arrays in element space or beamspace via unitary ESPRIT

Abstract: The UCA-ESPRIT is a closed-form algorithm developed for use in conjunction with a uniform circular array (UCA) that provides automatically paired source azimuth and elevation angle estimates. The 2-D unitary ESPRIT is presented as an algorithm providing the same capabilities for a uniform rectangular array (URA). In the final stage of the algorithm, the real and imaginary parts of the ith eigenvalue of a matrix are one-to-one related to the respective direction cosines of the ith source relative to the two major array axes. The 2-D unitary ESPRIT offers a number of advantages over other proposed ESPRIT based closed-form 2-D angle estimation techniques. First, except for the final eigenvalue decomposition of a dimension equal to the number of sources, it is efficiently formulated in terms of real-valued computation throughout. Second, it is amenable to efficient beamspace implementations that are presented. Third, it is applicable to array configurations that do not exhibit identical subarrays, e.g., two orthogonal linear arrays. Finally, the 2-D unitary ESPRIT easily handles sources having one member of the spatial frequency coordinate pair in common. Simulation results are presented verifying the efficacy of the method.

Small-Sample Bias in GMM Estimation of Covariance Structures

Abstract: We examine the small-sample properties of the generalized method of moments estimator applied to models of covariance structures, in which case it is commonly known as the optimal minimum distance (OMD) estimator. We find that OMD is almost always biased downward in absolute value. The bias arises because sampling errors in the second moments are correlated with sampling errors in the weighting matrix used by OMD. Furthermore, OMD is usually dominated by equally weighted minimum distance (EWMD). We also propose an alternative estimator that is unbiased and asymptotically equivalent to OMD. The Monte Carlo evidence indicates, however, that it is usually dominated by EWMD.

Simulation of Ergodic Multivariate Stochastic Processes

Abstract: A simulation algorithm is proposed to generate sample functions of a stationary, multivariate stochastic process according to its prescribed cross-spectral density matrix. If the components of the vector process correspond to different locations in space, then the process is nonhomogeneous in space. The ensemble cross-correlation matrix of the generated sample functions is identical to the corresponding target. The simulation algorithm generates ergodic sample functions in the sense that the temporal cross-correlation matrix of each and every generated sample function is identical to the corresponding target, when the length of the generated sample function is equal to one period (the generated sample functions are periodic). The proposed algorithm is based on an extension of the spectral representation method and is very efficient computationally since it takes advantage of the fast Fourier transform technique. The generated sample functions are Gaussian in the limit as the number of terms in the frequency discretization of the cross-spectral density matrix approaches infinity. An example involving simulation of turbulent wind velocity fluctuations is presented in order to demonstrate the capabilities and efficiency of the proposed algorithm.

Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs

Abstract: Abstract The matrix volume of coal shrinks when occluded gases desorb from its structure. In coalbed gas reservoirs, matrix shrinkage could cause the fracture aperture width to increase, causing an increase in permeability. A computer model was developed, based on elastic rock mechanics principles, to evaluate the potential effect of matrix shrinkage on the absolute permeability of coalbed reservoirs as fluid pressure is drawn down during gas production. The model predicts that the fracture width can potentially increase, depending on the combined influence of a number of parameters, particularly Young’s modulus of elasticity, Poisson’s ratio, fracture spacing and matrix shrinkage parameters. Each of these parameters vary depending on coal composition, so each individual coal will behave differently. A sensitivity study was conducted to evaluate the influence of each model parameter using a geologically reasonable range of input values. ‘Base case’, upper and lower limits were selected, based on published data. Gas production was simulated by reducing fluid pressure from 1290 PSI (8.89 MPa) to 100 PSI (0.70 MPa). The matrix shrinkage parameter εmax was found to produce the largest effect on permeability. Permeability changes as large as +250 mD are predicted for the upper case value of εmax. If εmax is small, however, the predicted permeability change will be negligible. An increase in Young’s modulus, Poisson’s ratio and fracture spacing each cause a predicted increase in permeability. Results of this model study should be verified through additional modelling, laboratory and field-based studies.

A Singularly Valuable Decomposition: The SVD of a Matrix

Abstract: (1996). A Singularly Valuable Decomposition: The SVD of a Matrix. The College Mathematics Journal: Vol. 27, No. 1, pp. 2-23.

Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems.

Abstract: We present a unified theoretical approach to electromagnetic plane waves reflected or transmitted at arbitrarily anisotropic and homogeneous layered systems. Analytic expressions for the eigenvalues for the four-wave components inside a randomly oriented anisotropic medium are reported explicitly. As well, the partial transfer matrix for a slab of a continuously twisted biaxial material at normal incidence is described. Transition matrices for the incident and exit media are presented. Hence, a complete analytical 4\ifmmode\times\else\texttimes\fi{}4 matrix algorithm is obtained using Berreman's 4\ifmmode\times\else\texttimes\fi{}4 differential matrices [D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972)]. The algorithm has a general approach for materials with linear optical response behavior and can be used immediately, for example, to analyze ellipsometric investigations. \textcopyright{} 1996 The American Physical Society.

Second-order complex random vectors and normal distributions

Abstract: Complex random vectors are usually described by their covariance matrix. This is insufficient for a complete description of second-order statistics, and another matrix called the relation matrix is necessary. Some of its properties are analyzed and used to express the probability density function of normal complex vectors. Various consequences are presented.

Object-orientd matrix programming using OX

Abstract: This programming manual accompanies the PcGive Professional 9.0 software, an interactive modelling package for modelling statistical and economic data for forecasting or observing trends.

Apparatus for playing bingo on a slot machine

Abstract: A method and apparatus for playing the game of bingo on a slot machine is disclosed. The bingo slot machine includes a display matrix capable of generating random numbers and a plurality of wheels, the number of wheel corresponding to the number of columns used in the display matrix, and a slot machine activating arm for use to activate the wheels as would be typically found in a slot machine. The game is played by pulling the arm and activating the wheels which stop at random wheel positions. The positions are then compared to the display matrix in a fashion similar to the game of bingo.

Universal quantum groups

Abstract: For each invertible m×m matrix Q a compact matrix quantum group Au(Q) is constructed. These quantum groups are shown to be universal in the sense that any compact matrix quantum group is a quantum subgroup of some of them. Their orthogonal version Ao(Q) is also constructed. Finally, we discuss related constructions in the literature.

Gutzwiller's Trace Formula and Spectral Statistics: Beyond the Diagonal Approximation.

Abstract: We calculate the 2-point spectral correlation function for classically chaotic systems in the semiclassical limit using Gutzwiller’s trace formula. The off-diagonal contributions from pairs of nonidentical periodic orbits are evaluated by relating them to the diagonal terms. The behavior we find is similar to that recently discovered to hold for disordered systems using nonperturbative supersymmetric methods. Our analysis generalizes immediately to include parametric statistics and higher-order correlations and to the study of the semiclassical distribution of matrix elements. [S0031-9007(96)00955-6]

Spin dynamics in magnets: Equation of motion and finite temperature effects.

Abstract: General equations of motion are introduced for the evaluation of spin dynamics in magnetic materials. The theory uses the adiabatic separation of diagonal and off-diagonal components of the spin density matrix. This adiabatic approach considers the orientation of the local magnetic moments to be slowly varying relative to their magnitudes. The angles of the magnetization density are introduced as collective variables in density functional theory. The equations and technique can be simultaneously combined with those of first-principles molecular dynamics for the consistent treatment of spin-lattice interactions. Stochastic and deterministic approaches for treating finite temperature effects are introduced for such dynamics. The method is implemented within the local density approximation and applied to \ensuremath{\gamma}-Fe, a frustrated system where we obtain additional low-energy magnetic configurations. \textcopyright{} 1996 The American Physical Society.

Matrix Market: a web resource for test matrix collections

Abstract: We describe a repository of data for the testing of numerical algorithms and mathematical software for matrix computations. The repository is designed to accommodate both dense and sparse matrices, as well as software to generate matrices. It has been seeded with the well known Harwell-Boeing sparse matrix collection. The raw data files have been augmented with an integrated World Wide Web interface which describes the matrices in the collection quantitatively and visually, For example, each matrix has a Web page which details its attributes, graphically depicts its sparsity pattern, and provides access to the matrix itself in several formats. In addition, a search mechanism is included which allows retrieval of matrices based on a variety of attributes, such as type and size, as well as through free-text search in abstracts. The URL is http://math.nist.gov/MatrixMarket.