Showing papers on "Matrix (mathematics) published in 1971"

The biplot graphic display of matrices with application to principal component analysis

Abstract: SUMMARY Any matrix of rank two can be displayed as a biplot which consists of a vector for each row and a vector for each column, chosen so that any element of the matrix is exactly the inner product of the vectors corresponding to its row and to its column. If a matrix is of higher rank, one may display it approximately by a biplot of a matrix of rank two which approximates the original matrix. The biplot provides a useful tool of data analysis and allows the visual appraisal of the structure of large data matrices. It is especially revealing in principal component analysis, where the biplot can show inter-unit distances and indicate clustering of units as well as display variances and correlations of the variables. Any matrix may be represented by a vector for each row and another vector for each column, so chosen that the elements of the matrix are the inner products of the vectors representing the corresponding rows and columns. This is conceptually helpful in understanding properties of matrices. When the matrix is of rank 2 or 3, or can be closely approximated by a matrix of such rank, the vectors may be plotted or modelled and the matrix representation inspected physically. This is of obvious practical interest for the analysis of large matrices. Any n x m matrix Y of rank r can be factorized as

Theory of characteristic modes for conducting bodies

Abstract: A theory of characteristic modes for conducting bodies is developed starting from the operator formulation for the current. The mode currents form a weighted orthogonal set over the conductor surface, and the mode fields form an orthogonal set over the sphere at infinity. It is shown that the modes are the same ones introduced by Garbacz to diagonalize the scattering matrix of the body. Formulas for the use of these modes in antenna and scatterer problems are given. For electrically small and intermediate size bodies, only a few modes are needed to characterize the electromagnetic behavior of the body.

Linear Canonical Transformations and Their Unitary Representations

Abstract: We show that the group of linear canonical transformations in a 2N‐dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well‐known dynamical group of the N‐dimensional harmonic oscillator. Finally, we study the case of n particles in a q‐dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)× O (q). An application to the calculation of matrix elements is given in a following paper.

Correlation problems in atomic and molecular systems III. Rederivation of the coupled-pair many-electron theory using the traditional quantum chemical methodst

Abstract: The equations of the coupled-pair many-electron theory (CPMET) for the closed shell systems are rederived both in the spin-orbital and orbital forms without the use of second quantization, Wick's theorem or the technique of Feynman-like diagrams Only the Slater rules are used for the calculation of necessary matrix elements A comparison with earlier papers shows clearly the usefulness and conceptual simplicity of the mathematical methods of quantum field theory both in the derivation of the CPMET, in spin-orbital form, and in the process of excluding spin variables

Projection method for solving a singular system of linear equations and its applications

Abstract: The iterative method for solving system of linear equations, due to Kaczmarz [2], is investigated. It is shown that the method works well for both singular and non-singular systems and it determines the affine space formed by the solutions if they exist. The method also provides an iterative procedure for computing a generalized inverse of a matrix.

Current Matrix Elements from a Relativistic Quark Model

Abstract: A relativistic equation to represent the symmetric quark model of hadrons with harmonic interaction is used to define and calculate matrix elements of vector and axial-vector currents. Elements between states with large mass differences are too big compared to experiment, so a factor whose functional form involves one arbitrary constant is introduced to compensate this. The vector elements are compared with experiments on photoelectric meson production, Kl3 decay, and ω→πγ. Pseudoscalar-meson decay widths of hadrons are calculated supposing the amplitude is proportional (with one new scale constant) to the divergence of the axial-vector current matrix elements. Starting only from these two constants, the slope of the Regge trajectories, and the masses of the particles, 75 matrix elements are calculated, of which more than 3/4 agree with the experimental values within 40%. The problems of extending this calculational scheme to a viable physical theory are discussed.

Status of noninteracting control

Abstract: The current status of decoupling theory for linear constant multivariable systems is described. The subject is treated in vector space terms and appropriate background concepts including invariant and controllability subspaces are discussed. Suggestions are given for translating vector space operations into matrix operations suitable for computation. The controllability subspace is used to formulate the restricted (static compensation) decoupling problem. Although the most general version of this problem is unsolved, there are known solutions for three special cases. A complete solution to the extended (dynamic compensation) decoupling problem is known. If a linear constant multivariable system can be decoupled at all, by any means whatever, then it can always be decoupled using linear dynamic compensation. The internal structure of a decoupled system is described in simple matrix terms. Using this representation, it is possible to characterize the system pole distributions which may be achieved while preserving a decoupled structure. A procedure is outlined for synthesizing a dynamic compensator of low order which will decouple a system. The procedure actually provides minimal order decoupling compensators for systems in which the number of open-loop inputs equal the number of outputs to be controlled.

The Sparse Tableau Approach to Network Analysis and Design

Abstract: The tableau approach to automated network design optimization via implicit, variable order, variable time-step integration, and adjoint sensitivity computation is described. In this approach, the only matrix operation required is that of repeatedly solving linear algebraic equations of fixed sparsity structure. Required partial derivatives and numerical integration is done at the branch level leading to a simple input language, complete generality and maximum sparsity of the characteristic coefficient matrix. The bulk of computation and program complexity is thus located in the sparse matrix routines; described herein are the routines OPTORD and 1-2-3 GNSO. These routines account for variability type of the matrix elements in producing a machine code for solution of Ax=b in nested iterations for which a weighted sum of total operations count and round-off error incurred in the optimization is minimized.

Optimal linear regulators: The discrete-time case

Abstract: In this paper the optimal discrete-time linear-quadratic regulator problem is carefully presented and the basic results are reviewed. Dynamic programming is used to determine the optimization equations. Special attention is given to problems unique to the discrete-time case; this includes, for example, the possibility of a singular system matrix and a singular control-effort weighting matrix. Some problems associated with sampled-data systems are also summarized, e.g., sensitivity to sampling time, and loss of controllability due to sampling. Computational methods for the solution of the optimization equations are outlined and a simple example is included to illustrate the various computational approaches.

Unified theory of linear estimation

Abstract: We consider a general Gauss-Markoff model (Y, Xβ, σ 2 V), where E(Y)=Xβ, D(Y)=σ 2 V There may be deficiency in R(X), the rank of X and V may be singular Two unified approaches to the problem of finding BLUE's (minimum variance linear unbiased estimators) have been suggested One is a direct approach where the problem of inference on the unknown β is reduced to the numerical evaluation of the inverse of a partitioned matrix The second is an analogue of least squares, where the matrix used in defining the quadratic form in (Y-Xβ) to be minimized is a g-inverse of (V+XUX') in all situations, whether V is nonsingular or not, where U is arbitrary subject to a condition Complete robustness of BLUE's under different alternatives for V has been examined A study of BLE's (minimum mean square estimators) without demanding unbiasedness is initiated and a case has been made for further examination The unified approach is made possible through recent advances in the calculus of generalized inverse of matrices (see the recent book by Rao and Mitra, 1971a)

The Design of Multistage Separable Planar Filters

Abstract: A two-dimensional, or planar, digital filter can be described in terms of its planar response function, which is in the form of a matrix of weighting coefficients, or filter array. In many instances the dimensions of these matrices are so large that their implementation as ordinary planar convolutional filters becomes computationally inefficient. It is possible to expand the given coefficient matrix into a finite and convergent sum of matrix-valued stages. Each stage can be separated with no error into the product of an m-length column vector multiplied into an n-length row vector, where m is the number of rows and n is the number of columns of the original filter array. Substantial savings in computer storage and speed result if the given filter array can be represented with a tolerably small error by the first few stages of the expansion. Since each constituent stage consists of two vector-valued factors, further computational economies accrue if the one-dimensional sequences described by these vectors are in turn approximated by one-dimensional recursive filters. Two geophysical examples have been selected to illustrate how the present design techniques may be reduced to practice.

An Empirical Application and Evaluation of Discrete Stochastic Programming in Farm Management

Abstract: Discrete stochastic programming has been suggested as a means of solving sequential decision problems under uncertainty, but as yet little or no empirical evidence of the capabilities of this technique in solving such problems has appeared. This paper presents in some detail an empirical application of discrete stochastic programming, including a discussion of data requirements, matrix construction, and solution interpretation. Based on this empirical evidence, the problem-solving potential of the technique is evaluated.

Further contributions to the theory of generalized inverse of matrices and its applications

Abstract: This is a sequel to an earlier paper by the authors on the same subject presented at the Sixth Berkeley Symposium. In the previous paper, the authors have discussed there basic types of g-inverses-the minimum norm g-inverse, the least squares g-inverse and the minimum norm least squares g-inverse. In the paper these concepts are extended to more general situations involving semi norms in place of norms used earlier. It shown that a matrix is uniquely determined by its class of g-inverses. Further the subclass of g-inverses with a specified rank is characterized. Partial isometrics are discussed in a general set-up with reference to a pair linear spaces furnished with arbitrary quadratic norms. A unified theory of linear estimation is presented using the expression for a minimum semi norm inverse.

Computer field-matching solution of waveguide transverse discontinuities

Abstract: A computational method for solving a wide range of transverse waveguide discontinuity problems is described. Results are obtained by the simultaneous solution of matrix equations, generated by Fourier analysis, which relate the complex amplitudes of orthogonal electric and magnetic field components. In some cases, the solution is found to be sensitive to the way in which infinite series of field functions are truncated, and it is shown how the optimum form of truncation can be determined for many configurations of practical importance. Several examples showing the application of the method are given, and comparison of results with those obtained by experiment, and by other analytical techniques, confirms its accuracy.

Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm

Abstract: The theory of dilute polymer solutions based on the bead/spring model in the form given by Zimm is reformulated for arbitrary homogeneous flow histories. It is shown that the center of resistance of a polymer molecule moves with the solvent. By a preliminary transformation [3.4], the equations for the center-of- resistance motion are separated from those for the motion of the N spring vectors. The spring-vector equations involve a symmetric non-singular matrix B [3.14] whose characteristic values equal the non-zero characteristic values of Zimms singular matrix HA. A further transformation [4.2] which diagonalizes B yields separate differential eq. [5.4] for pq*, the polymer contribution to the stress tensor associated with the q normal mode. Transformation to an embedded basis enables one to integrate these equations so as to obtain pq* in terms of the flow history ([5.6], [5.9]), and summation over q then gives the required constitutive eq. [5.17] for the polymer solution. These are of the same form as the “rubber-like liquid” constitutive equations (with addition of a solvent-contribution term) derived from the network theory of Lodge, but the memory function is determined to within three constants (e. g. N, h*, τ1). Peterlina solution for the normal-coordinate distribution function in steady shear flow is generalized for an arbitrary homogeneous (time-dependent or steady) flow and expressed in terms of pq* which can be evaluated when the flow history is given.

Dispersion characteristics of slot lines

Abstract: A new numerical method is developed for the analysis of dispersion characteristics of slot lines. Galerkin's method is applied in the Fourier-transform domain to derive a determinantal equation for the propagation constants. It is shown that accurate numerical results can be obtained with even a 2 × 2 matrix.

An Iterative Algorithm for Computing the Best Estimate of an Orthogonal Matrix

Abstract: The closest unitary matrix, measured in Euclidean norm, to a given rectangular matrix A is known to be the unitary factor in the polar decomposition of A The paper gives a family of iterative methods of order of convergence $p + 1,\, p = 1,2,3, \cdots $, for computing this matrix The methods are especially efficient when the columns of A are not too far from being orthonormal The choice of order of convergence to minimize the amount of computation is discussed Global convergence properties for the methods of order $ \leqq 4$ are studied and sufficient conditions for convergence in terms of $\| {I - A^H A} \|$ are given

On some desirable patterns in block designs

Abstract: SUMMARY A general iterative formula for the covariance matrix of the adjusted treatment means in block designs is given and its relation to design patterns discussed. It is shown that the general formula becomes very simple if the design has some degree of balance. Particularly interesting are those designs that combine features of balance and orthogonal designs. A matrix derived in a simple way from the incidence matrix proves useful in describing some properties of block designs. The simplification of the general formula depends entirely on the pattern of this matrix which also determines the efficiency of the design. Furthermore, its particular relation to treatment contrasts is helpful in designing block experiments of desirable properties. The construction of some useful designs that are simple in analysis and practical in application is illustrated by several examples.

Ultraviolet Absorption Spectra of Transition Metal Atoms in Rare-Gas Matrices.

Abstract: Absorption spectra from 2200 to 4000 A have been obtained of chromium, manganese, iron, cobalt, copper, nickel, tin, and palladium atoms trapped in argon matrices at 4.2°K and of iron and copper in krypton and xenon matrices at 4.2 and 20°K. Observed transitions were found to correlate with gas phase transitions, under the assumption of a matrix and atomic configuration dependent shift of the atomic transitions. Energy shifts of the transitions were inversely proportional to matrix atom size. Configurations with an odd number of 3d electrons were shifted less than those with an even number. Within a given configuration, the transitions at higher energies were shifted more than those at lower energies. A Lennard‐Jones potential was unsuccessful in generating the observed energy shifts caused by the interaction between the trapped atom and the matrix. Atom diffusion and resultant aggregation within the matrix both during the condensation of the solid from the gas phase and during warming was found to be a s...

Application of the Theory of Irreducible Tensor Operators to Molecular Hyperfine Structure

Abstract: The theory of irreducible tensor operators is employed to evaluate the matrix elements of the electric quadrupole and magnetic dipole interactions for molecules with several coupling nuclei Two different coupling schemes are considered, namely the unequal coupling and nearly equal coupling representations The relative intensities of the hyperfine transitions are also derived Application of the theory is stressed in an expository manner

Finite-Element Analysis of Magnetically Saturated D-C Machines

Abstract: The magnetic field distribution in saturated iron parts of electric machines is defined by a nonlinear quasi-Poisson equation. Solution of this equation is equivalent to minimization of a nonlinear energy functional. A recent paper has proposed approximate minimization by means of a finite element method, using triangular finite elements and a quadratically convergent iteration scheme. This new method is now applied to a 5 KW d-c machine, whose no-load and on-load characteristics are predicted and compared with experimental measurements. Good agreement is obtained. Since the pole axis is not an axis of magnetic symmetry under load, a periodicity condition is introduced to relate all magnetic vector potentials to those one pole pitch away. This condition is enforced by means of a special connection matrix, whose derivation is shown in the paper. An automatic plotting program has been developed for graphical plotting of the flux distributions, and several field plots for the machine are shown.

Canonical Transformations and Matrix Elements

Abstract: We use the ideas on linear canonical transformations developed previously to calculate the matrix elements of the multipole operators between single‐particle states in a three‐dimensional oscillator potential. We characterize first the oscillator states in the chain of groups Sp(6)⊃Sp(2)×O(3), Sp(2)⊃OS(2), and O(3)⊃OL(2), and then expand the multipole operators in terms of irreducible tensors with respect to the Sp(2)×O(3) group. Their matrix elements are obtained by applying the Wigner‐Eckart theorem with respect to both the Sp(2) and O(3) groups. In this way an explicit expression for the radial integral of rk, k > 0, is obtained.

Minimization Algorithms Making Use of Non-quadratic Properties of the Objective Function

Abstract: This paper proposes modifications to the matrix updating formulae used in the FletcherPowell (1963), Broyden (1970) and Fletcher (1970) methods for function minimization, and discusses some properties of these modified formulae. A function minimization algorithm incorporating the new expressions has been programmed and the results of tests with some well-known functions are reported.

"Inners" approach to some problems of system theory

Abstract: In this paper a formal definition of "inners" of a square matrix is introduced. Also defined are the concepts of positive, negative, null, and nonnull innerwise matrix. Applications of these definitions are shown 1) for the necessary and sufficient conditions for the roots of a real polynomial to be within the unit circle, 2) for the roots of a real polynomial to be distinct and on the real axis, and 3) to be distinct and on the imaginary axis in the complex plane. Conditions on relative stability of linear continuous systems as well as on root distribution within the unit circle are also presented in terms of inners. Furthermore, a theorem establishing the equivalence between positive innerwise matrix and a positive definite symmetric matrix is derived for the case when all the roots of a real polynomial are inside the unit circle. Extension of the inners approach to the general problems of clustering of roots as well as of the root distribution in the complex plane is briefly mentioned. Several examples are presented to elucidate the application of the inners concept to some problems that arise in system theory.