Showing papers on "Matrix (mathematics) published in 1975"
TL;DR: In this paper, the linear time-discrete state-space model is generalized from single-dimensional time to two-dimensional space, which includes extending certain basic known concepts from one to two dimensions, such as the general response formula, state transition matrix, Cayley-Hamilton theorem, observability, and controllability.
Abstract: The linear time-discrete state-space model is generalized from single-dimensional time to two-dimensional space. The generalization includes extending certain basic known concepts from one to two dimensions. These concepts include the general response formula, state-transition matrix, Cayley-Hamilton theorem, observability, and controllability.
1,710 citations
TL;DR: It is shown how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization.
Abstract: A minimal basis of a vector space V of n-tuples of rational functions is defined as a polynomial basis such that the sum of the degrees of the basis n-tuples is minimum. Conditions for a matrix G to represent a minimal basis are derived. By imposing additional conditions on G we arrive at a minimal basis for V that is unique. We show how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization. Transfer function matrices G solving equations of the form $PG = Q$ are also obtained by this method; applications to the problem of finding minimal order inverse systems are given. Previous applications to convolutional coding theory are noted. This range of applications suggests that minimal basis ideas will be useful throughout the theory of multivariable linear systems. A restatement of these ideas in the language of valuation theory is given in an Ap...
743 citations
TL;DR: In this article, the problem of estimating the regression coefficient matrix having known (reduced) rank for the multivariate linear model when both sets of variates are jointly stochastic is discussed.
548 citations
TL;DR: For a matrix A which is diagonally dominant both by rows and by columns, bounds are given which can be used to give a lower bound for the smallest singular value.
361 citations
TL;DR: For an m-state homogeneous Markov chain whose one-step transition matrix is T, the group inverse, $A^#$, of the matrix $A = I - T$ is shown to play a central role as discussed by the authors.
Abstract: For an m-state homogeneous Markov chain whose one-step transition matrix is T, the group inverse, $A^#$, of the matrix $A = I - T$ is shown to play a central role. For an ergodic chain, it is demon...
354 citations
TL;DR: Several of the more stable methods for computing the Jordan canonical form are discussed together with the alternative approach of computing well-defined bases (usually orthogonal) of the relevant invariant subspaces.
Abstract: The solution of the complete eigenvalue problem for a non-normal matrix A presents severe practical difficulties when A is defective or close to a defective matrix. However in the presence of rounding errors one cannot even determine whether or not a matrix is defective. Several of the more stable methods for computing the Jordan canonical form are discussed together with the alternative approach of computing well-defined bases (usually orthogonal) of the relevant invariant subspaces.
266 citations
TL;DR: A new method of reduction in order of linear time-invariant systems is introduced that combines the desirable features of the time-moments and the modal methods of reduction based on the concept of Pade approximation about more than one point.
Abstract: A new method of reduction in order of linear time-invariant systems is introduced. It combines the desirable features of the time-moments and the modal methods of reduction. The methods is based on the concept of Pade approximation about more than one point. The reduced model derived by this method, retains the dominant modes (or any desirable modes) and fits the initial time-moments of the system The method is extended to the problem of reducing a high-order multivariable system described by its matrix transfer function. Several illustrative examples are discussed throughout the paper.
218 citations
TL;DR: The J-matrix method introduced previously for s-wave scattering is extended to treat the lth partial wave kinetic energy and Coulomb Hamiltonians within the context of square integrable (L2), Laguerre (Slater), and oscillator (Gaussian) basis sets.
Abstract: The J-matrix method introduced previously for s-wave scattering is extended to treat the lth partial wave kinetic energy and Coulomb Hamiltonians within the context of square integrable (L2), Laguerre (Slater), and oscillator (Gaussian) basis sets The determination of the expansion coefficients of the continuum eigenfunctions in terms of the L2 basis set is shown to be equivalent to the solution of a linear second order differential equation with appropriate boundary conditions, and complete solutions are presented Physical scattering problems are approximated by a well-defined model which is then solved exactly In this manner, the generalization presented here treats the scattering of particles by neutral and charged systems The appropriate formalism for treating many channel problems where target states of differing angular momentum are coupled is spelled out in detail The method involves the evaluation of only L2 matrix elements and finite matrix operations, yielding elastic and inelastic scattering information over a continuous range of energies
173 citations
TL;DR: In this article, the authors show that the transition from the origially given statistical scattering matrix S to the matrix S = USU T reduces the Hauser-Feshbach problem with direct reactions to the one without.
170 citations
TL;DR: The problem of determining the nonlinear function (“blackbox”) which optimally associates two sets of data is considered and an iteration method based on the concept of the generalized inverse of a matrix provides the polynomial mapping of degreek onX by whichY is retrieved in an optimal way in the least squares sense.
Abstract: The problem of determining the nonlinear function (“blackbox”) which optimally associates (on given criteria) two sets of data is considered. The data are given as discrete, finite column vectors, forming two matricesX (“input”) andY (“output”) with the same numbers of columns and an arbitrary numbers of rows. An iteration method based on the concept of the generalized inverse of a matrix provides the polynomial mapping of degreek onX by whichY is retrieved in an optimal way in the least squares sense. The results can be applied to a wide class of problems since such polynomial mappings may approximate any continuous real function from the “input” space to the “output” space to any required degree of accuracy. Conditions under which the optimal estimate is linear are given. Linear transformations on the input key-vectors and analogies with the “whitening” approach are also discussed. Conditions of “stationarity” on the processes of whichX andY are assumed to represent a set of sample sequences can be easily introduced. The optimal linear estimate is given by a discrete counterpart of the Wiener-Hopf equation and, if the key-signals are noise-like, the holographic-like scheme of associative memory is obtained, as the optimal nonlinear estimator. The theory can be applied to the system identification problem. It is finally suggested that the results outlined here may be relevant to the construction of models of associative, distributed memory.
TL;DR: The Perron-Frobenius theory for square, irreducible, nonnegative matrices is generalized by studying the structure of the algebraic eigenspace of an arbitrary square nonnegative matrix corresponding to its spectral radius.
TL;DR: In this paper, a simple and rapid technique for finding this inverse directly from a list of sires and dams, bypassing computation of the matrix whose inverse is wanted, is presented, which can be used to evaluate all relationships among animals to be evaluated.
TL;DR: In this article, the extreme rays of complex and real diagonally dominant matrices and their duals are identified and several results on lattices of faces of cones are given.
Abstract: The extreme rays of several cones of complex and real diagonally dominant matrices, and their duals, are identified Several results on lattices of faces of cones are given It is then shown that the dual (in the real space of hermitian matrices) of the cone of hermitian diagonally dominant matrices cannot be the image of the cone of positive semidefinite matrices under any nonsingular linear transformation; in particular, it cannot be the image of the cone of positive semidefinite matrices under the Ljapunov transformation ΈH AH+ HA* determined by a positive stable matrix A
TL;DR: In this paper, the problem of incompressible potential flow about two-and three-dimensional bodies is formulated as an integral equation over the body surface, and the solution can be calculated on the surface without considering the remainder of the flow field.
IBM1
TL;DR: In this paper, the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix is derived and a convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the original matrix is held constant.
Abstract: Properties of the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix are derived Such a sum is convex but not necessarily everywhere differentiable A convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the matrix is held constant An implementation of this procedure is described and numerical results are included
TL;DR: In this article, the differential equations governing the propagation in time of the sensitivity matrix for a mathematical model given by a system of ordinary differential equations are derived and used to perform a statistical sensitivity analysis of models for chemical reactors.
Abstract: The differential equations governing the propagation in time of the sensitivity matrix for a mathematical model given by a system of ordinary differential equations are derived. These equations are used to perform a statistical sensitivity analysis of models for chemical reactors. The behavior of the sensitivities at equilibrium is analyzed. It is shown that the sensitivity equations for linear kinetics may be solved using an analytic representation. The numerical solution of these equations is discussed, and illustrative examples are presented. The lognormal distribution is presented as being representative of errors in rate constants.
TL;DR: In this paper, a differential matrix eigenvalue problem is used to generate systems of nonlinear evolution equations, including triad, multitriad, self-modal, and quartet wave interactions.
Abstract: A differential matrix eigenvalue problem is used to generate systems of nonlinear evolution equations. They model triad, multitriad, self‐modal, and quartet wave interactions. A nonlinear string equation is also recovered as a special case. A continuum limit of the eigenvalue problem and associated evolution equations are discussed. The initial value solution requires an investigation of the corresponding inverse‐scattering problem.
TL;DR: In this article, a method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section.
Abstract: A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.
TL;DR: In this article, the method of moments is applied to the computation of the charge distributions and capacitance matrix for electrostatic systems of bare and dielectric-coated cylindrical wires.
Abstract: The method of moments is applied to the computation of the charge distributions and capacitance matrix for electrostatic systems of bare and dielectric-coated cylindrical wires. Several choices of expansion functions are investigated in detail and compared. Harmonic series expansion functions are shown to be especially well suited to problems involving systems of closely-spaced dielectric-coated cylindrical wires.
TL;DR: In this paper, the authors compared the spin relaxation theories of Albers and Deutch and of Argyres and Kelley based on different projection operator methods, and the theory of Freed based on generalized cumulant expansions and showed that the first two yield equivalent expressions for the time evolution of the spin density matrix.
Abstract: The general spin‐relaxation theories of Albers and Deutch and of Argyres and Kelley based on different projection operator methods, and the theory of Freed based on generalized cumulant expansions are compared. It is shown that the first two yield equivalent expressions for the time evolution of the spin density matrix. They are also found to be equivalent to a cumulant expansion based on total ordering of the cumulant operators (TTOC), which is different from the partial time ordering method (PTOC) used by Freed. The TTOC method is found to be the more convenient for the frequency domain (i.e., for calculating spectra), while the PTOC method is for time domain analyses. Examples of the use of the TTOC method are given. Useful expressions are given for the case where the lattice may be treated in terms of classical Markov processes, but, in general, it is found that for such cases the stochastic Liouville method is the more useful for computations.
TL;DR: For polynomial splines this matrix is closely related to an oscillation matrix and its eigenvectors show the typical sign distribution as discussed by the authors, which is the basis for a variant of spline smoothing.
Abstract: Spline smoothing can be reduced to the minimization of a certain quadratic form with positive semidefinite matrix. For polynomial splines this matrix is closely related to an oscillation matrix and its eigenvectors show the typical sign distribution. This fact is the basis for a variant of spline smoothing.
TL;DR: In this article, a variational method for the two-body density matrix is developed for practical calculations of the properties of many-fermion systems with two−body interactions, in which the energy E = JHijkl ρijkl is minimized using the two−Body density matrix elements ρjkl = 〈ψ‖a+ja+iakal‖ψ〉 as variational parameters.
Abstract: A variational method for the two−body density matrix is developed for practical calculations of the properties of many−fermion systems with two−body interactions. In this method the energy E = JHijkl ρijkl is minimized using the two−body density matrix elements ρijkl = 〈ψ‖a+ja+iakal‖ψ〉 as variational parameters. The approximation consists in satisfying only a subset of necessary conditions—the nonnegativity of the following matrices: the two−body density matrix, the ’’two−hole matrix’’ Qijkl = 〈Ψ‖ajaia+ka+l‖Ψ〉 and the particle−hole matrix Gijkl = 〈Ψ‖ (a+iaj−ρij)+ (a+kal−ρk) ‖Ψ〉. The idea of the method was introduced earlier; here some further physical interpretation is given and a numerical procedure for calculations within a small single−particle model space is described. The method is illustrated on the ground state of Be atom using 1s, 2s, 2p orbitals.
TL;DR: In this paper, the inverse of a large numerator relationship matrix is used to compute the relationship matrix with little additional labor and the number of equations for groups to account for genetic trend and for differences among subpopulations is reduced.
Patent•
01 Dec 1975
TL;DR: A sustained release pharmaceutical composition which includes a pharmacological material, a biological binding agent for the pharmacological materials, and a matrix of a water-insoluble but water-swellable hydrophilic polymer is presented in this paper.
Abstract: A sustained release pharmaceutical composition which includes a pharmacological material; a biological binding agent for the pharmacological material; and a matrix of a water-insoluble but water-swellable hydrophilic polymer.
TL;DR: In this paper, the most important identifiable relationships among sires used in artificial insemination are due to common sires and maternal grandsires and a rapid method for finding the inverse of a numerator relationship matrix resulting from use of these two points in the pedigree has been developed.
TL;DR: In this paper, the authors derived explicit matrix formulae for the calculation of total exchange areas in the context of Hottel's zone method and obtained working relations for the case of a uniform grey absorbing-emitting/isotropically-scattering medium confined in a Lambert enclosure.
TL;DR: Fortran IV subroutines for the in-core solution of linear algebraic systems with a sparse, symmetric, skyline-stored coefficient matrix are presented and the application to ‘superelement’ condensation of large-scale systems is discussed.
TL;DR: In this paper, the authors studied the convergence of the linear stationary iterative process (LPI) with respect to a nonsingular matrix, and provided convergence criteria for the singular case.
Abstract: For a square, possibly singular, matrix A decomposed as $A = M - N$ where M is nonsingular, let $T = M^{ - 1} N$. The Drazin inverse of $I - T$ is used to review well-known conditions under which the powers of T converge to some matrix. These concepts are then applied to the study of the convergence of the linear stationary iterative process $x^{(k + 1)} = Tx^{(k)} + M^{ - 1} b$, which is used to approximate solutions to consistent linear systems $Ax = b$ . When the process converges, the limit is given in terms of the Drazin inverse of $I - T$ and asymptotic rates of convergence are discussed. The concept of a regular splitting of a nonsingular matrix is extended to the singular case in a natural way and convergence criteria are established. Finally, it is shown that a matrix A has a regular splitting $A = M - N$ such that the powers of $T = M^{ - 1} N$ converge if and only if $A = AXA$ is solvable for some nonsingular $X \geqq 0$, thus providing a complete extension of Varga’s characterization of a conv...