Showing papers on "Matrix (mathematics) published in 1978"
TL;DR: In this article, the rank estimation of the rank A of the matrix Y, i.e., the estimation of how much of the data y ik is signal and how much is noise, is considered.
Abstract: By means of factor analysis (FA) or principal components analysis (PCA) a matrix Y with the elements y ik is approximated by the model Here the parameters α, β and θ express the systematic part of the data yik, “signal,” and the residuals ∊ ik express the “random” part, “noise.” When applying FA or PCA to a matrix of real data obtained, for example, by characterizing N chemical mixtures by M measured variables, one major problem is the estimation of the rank A of the matrix Y, i.e. the estimation of how much of the data y ik is “signal” and how much is “noise.” Cross validation can be used to approach this problem. The matrix Y is partitioned and the rank A is determined so as to maximize the predictive properties of model (I) when the parameters are estimated on one part of the matrix Y and the prediction tested on another part of the matrix Y.
2,468 citations
TL;DR: In this article, a review of the algebras related to Kronecker products is presented, which have several applications in system theory including the analysis of stochastic steady state.
Abstract: The paper begins with a review of the algebras related to Kronecker products. These algebras have several applications in system theory including the analysis of stochastic steady state. The calculus of matrix valued functions of matrices is reviewed in the second part of the paper. This calculus is then used to develop an interesting new method for the identifiication of parameters of lnear time-invariant system models.
1,944 citations
TL;DR: In this article, the exponential of a matrix could be computed in many ways, including approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial.
Abstract: In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...
1,849 citations
TL;DR: A new iterative method for the solution of systems of linear equations has been recently proposed by Meijerink and van der Vorst and has been applied to real laser fusion problems taken from typical runs of the laser fusion simulation code LASNEX.
908 citations
01 Jan 1978
TL;DR: The given theory helps to explain the excellent numerical results that are obtained by a recent algorithm (Powell, 1977) by regarding the positive definite matrix that is revised on each iteration as an approximation to the second derivative matrix of the Lagrangian function.
Abstract: Variable metric methods for unconstrained optimization calculations can be extended to the constrained case by regarding the positive definite matrix that is revised on each iteration as an approximation to the second derivative matrix of the Lagrangian function. Linear approximations to the constraints are used. Han (1976) has analyzed the convergence of these methods in the case when the true second derivative matrix of the Lagrangian function is positive definite at the solution. However, this matrix sometimes has negative eigenvalues so we analyze the rate of convergence in this case. We find that it is still superlinear. Therefore we may continue to use positive definite second derivative approximations and there is no need to introduce any penalty terms. The given theory helps to explain the excellent numerical results that are obtained by a recent algorithm (Powell, 1977).
534 citations
TL;DR: In this article, a system of equations which must be satisfied by multiparticle matrix elements of any local operator in field theories with soliton behaviour is derived, and the form factors of various operators of interest are calculated exactly by means of the known exact S-matrices in the sine-Gordon, massive Thirring, nonlinear σ−, and Gross-Neveu models.
436 citations
TL;DR: The Lagrange function for the stiffness matrix weighted norm of the errors between the given and the optimal stiffness matrix unity matrix is defined in this paper, where the error is defined as the difference between the error between the desired stiffness matrix and the given stiffness matrix.
Abstract: Nomenclature Lagrange function for the flexibility matrix weighted norm of the errors between the given and the optimal flexibility matrix Lagrange function for the stiffness matrix weighted norm of the errors between the given and the optimal stiffness matrix unity matrix given stiffness matrix mass matrix M» //element of TV //element of N~* Nq general-coordinates vector measured mode shape /th measured_mode shape normalized 7} transpose of [ • ] = optimal flexibility matrix = (/ element of W orthogonal mode shape matrix = (/ element of X optimal stiffness matrix -ij element of Y = matrices of Lagrange multipliers = ij element of 0y and 0W , respectively = matrix of Lagrange multipliers = given flexibility matrix = matrices of Lagrange multipliers = ij element of A^ and A^ , respectively = measured frequency matrix = //element of Q y»(i* w
303 citations
TL;DR: In this paper, a factorized total S -matrix in two space-time dimensions with isotopic O(n) symmetry was constructed, and it was shown that this S-matrix is the exact one of the O( n ) chiral field.
302 citations
TL;DR: Hadamard matrices have been widely studied in the literature and many of their applications can be found in this paper, e.g., incomplete block designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III (SRSIII), optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects.
Abstract: An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications.
288 citations
TL;DR: In this article, a linear-time algorithm for sparse symmetric matrices which converts a matrix into pentadiagonal form (bandwidth 2) whenever it is possible to do so using simultaneous row and column permutations is presented.
Abstract: We present a linear-time algorithm for sparse symmetric matrices which converts a matrix into pentadiagonal form (“bandwidth 2”), whenever it is possible to do so using simultaneous row and column permutations. On the other hand when an arbitrary integer k and graph G are given, we show that it is $NP$-complete to determine whether or not there exists an ordering of the vertices such that the adjacency matrix has bandwidth $ \leqq k$, even when G is restircted to the class of free trees with all vertices of degree $ \leqq 3$. Related problems for acyclic directed graphs (upper triangular matrices) are also discussed.
287 citations
TL;DR: In this paper, it was shown that in the limit N→∞ integrals with respect to Haar measure of products of the elements of a matrix in SO(N) approach corresponding moments of a set of independent Gaussian random variables.
Abstract: We show that in the limit N→∞ integrals with respect to Haar measure of products of the elements of a matrix in SO(N) approach corresponding moments of a set of independent Gaussian random variables Similar asymptotic forms are obtained for SU(N) and Sp(N) An application of these results to Wilson’s formulation of lattice gauge theory is briefly considered
TL;DR: In this article, the probability that the estimated between-group covariance matrix is not positive deJinite is computed for the balanced single classlfication multivariate analysis of variance with random effects.
Abstract: The probability (Q) that the estimated between-group covariance matrix is not positive deJinite is computed for the balanced single classlfication multivariate analysis of variance with random effects. It is shown that Q depends only on the roots of the matrix product of the inverse of the true within-group and the true between-group covariance matrices which, for independent variables, reduces to expressions in intra-class correlations. Values of Q are computedfor ranges of size of experiment, intra-class correlation and number of variables. Even for large experiments, Q can approach 100% if there are many variables, for example with 160 groups of size 10 and either 8 independent variables each with intra-class 0.025 or 14 variables each with intra-class correlation 0.0625. Some rationalization of the results is given in terms of the bias in the roots of the sample between-group covariance matrix. In genetic applications, the between-group covariance matrix is proportional to the genetic covariance matrix, if non-positive definite, heritabilities and ordinary or partial genetic correlations are outside their valid limits, and the effiect on selection index construction is discussed.
TL;DR: In this article, the authors present new methods for computing the greatest common right divisor of polynomial matrices, which involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices.
Abstract: We present new methods for computing the greatest common right divisor of polynomial matrices. These methods involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices, which require no polynomial operations. They can provide a row proper greatest common right divisor, test for coprimeness and calculate dual dynamical indices. The generalized resultant matrices are developments of the scalar Sylvester and Bezoutian resultants and many of the familiar properties of these latter matrices are demonstrated to have analogs with the properties of the generalized resultant matrices for matrix polynomials.
TL;DR: Measurements have been made to determine all sixteen elements of the Mueller scattering matrix for two types of nonspherical particles, and the expected eight of the sixteen elements were found to be zero within measurement accuracy.
Abstract: Measurements have been made to determine all sixteen elements of the Mueller scattering matrix for two types of nonspherical particles. Rounded particles of ammonium sulfate and nearly cubic particles of sodium chloride in the 0.1–1.0-μm size range have been prepared by nebulizing salt water solutions and drying the droplets. Scanning electron micrographs are used to determine size distributions used in Mie calculations of all matrix elements. The expected symmetry of the scattering matrices across the diagonal was confirmed, and the expected eight of the sixteen elements were found to be zero within measurement accuracy. The rounded particles were found accurately to obey Mie theory, while the cubic particles were poorly described by Mie theory for some matrix elements and some angles. Total intensity and linear polarization measurements are presented also for a series of increasing sizes of rounded and cubic particles. A discussion of the effect of nonsphericity on the various matrix elements is given, and applications of these results are given to analysis of particle properties in the laboratory, the clouds of Venus, reflection nebulae, the zodiacal light, and atmospheric particulates.
TL;DR: In this article, a scattering matrix approach, that involves only the transition matrix of a single obstacle, is proposed for studying the multiple scattering of elastic waves in a medium (matrix) containing identical, long, parallel, randomly distributed cylinders of arbitrary cross section.
Abstract: A scattering matrix approach, that involves only the transition matrix of a single obstacle, is proposed for studying the multiple scattering of elastic waves in a medium (matrix) containing identical, long, parallel, randomly distributed cylinders of arbitrary cross section The elastic properties of the cylinders are assumed to be different from those of the matrix A statistical approach in conjunction with Lax’s ’’quasicrystalline’’ approximation is employed to obtain equations for the average amplitudes of the scattered and exciting fields which may then be solved to yield the dispersion relations of the composite medium Dynamic elastic properties of the composite medium containing circular and elliptical cylinders are found in the Rayleigh or low‐frequency limit Numerical results displaying phase velocity and damping effect of the composite medium are presented for a wide range of frequencies
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TL;DR: In the proof, it is shown two n - k degree homogeneous polynomials in n variables are equal by applying induction to those terms lacking one variable.
01 Jan 1978
TL;DR: In this paper, the use of plane-wave spectra for the representation of fields in space and the consideration of antenna-antenna (antenna-scatterer) interactions at arbitrary separation distances is discussed.
Abstract: From Abstract: "This monograph is distinguished by the use of plane-wave spectra for the representation of fields in space and by the consideration of antenna-antenna (antenna-scatterer) interactions at arbitrary separation distances." From Preface: "The primary objective of this monograph is to facilitate the critical acceptance and proper application of antenna and field measurement techniques deriving more or less directly from the plane-wave scattering matrix (PWSM) theory of antennas and antenna-antenna interactions. A second objective is to present some recent and some new theoretical results based on this theory."
01 Jan 1978
TL;DR: Formalization and extension of a recent proposal for adjusting initial unbalanced estimates of components of a matrix allows them to satisfy accounting requirements imposed by tabular form in an optimal manner, and the optimal adjustment of very large social account matrices becomes quite feasible.
Abstract: Formalization and extension of a recent proposal for adjusting initial unbalanced estimates of components of a matrix allows them to satisfy accounting requirements imposed by tabular form in an optimal manner. The proposal, which is based on linear combinations of initial unbiased estimates, has many potential applications in national income accounting, input-output construction, and demography, among other fields. Given that the adjustment procedure simply represents the first-order conditions resulting from the minimization of a quadratic loss function, it is possible to develop alternative procedures for minimizing the constrained loss function. These procedures, based on the conjugate gradient algorithm, prove to be much more efficient than the traditional solution, both in terms of the time taken and storage requirements, and the optimal adjustment of very large social account matrices becomes quite feasible. Application of these techniques to a social account matrix constructed for the Muda River District in West Malaysia indicate the feasibility and usefulness of the method. Statistical data are included.
TL;DR: A new derivation of a set, of complete invariants and a corresponding canonical form first given by Morse is provided, which yields relatively simple proofs and economical matrix algorithms.
Abstract: The class {∑} of all linear multivariable systems is partitioned into equivalence classes by the group consisting of all basis, all state feedback and all output injection transformations. This paper provides a new derivation of a set, of complete invariants and a corresponding canonical form first given by Morse. Strong reachability and strong observability concepts are the key tools. The method yields relatively simple proofs and economical matrix algorithms.
01 Jun 1978
TL;DR: In this article, a quaternion is regarded as a four-parameter representation of a coordinate transformation matrix, where the four components of the quaternions are treated on an equal basis.
Abstract: A quaternion is regarded as a four-parameter representation of a coordinate transformation matrix, where the four components of the quaternion are treated on an equal basis. This leads to a unified, compact, and singularity-free approach to determining the quaternion when the matrix is given.
Book•
01 Jan 1978
TL;DR: In this article, the authors propose linear systems of Equations and Generalized Inverses of Matrices and Matrix Differentiation with Constant Coefficients (GIC) with constant coefficients.
Abstract: 1. Vectors and Vector Spaces.- 2. Matrix Algebra.- 3. Linear Systems of Equations and Generalized Inverses of Matrices.- 4. Vectorization of Matrices and Matrix Functions: Matrix Differentiation.- 5. Systems of Difference Equations with Constant Coefficients.
TL;DR: A family of methods of Implicit Runge-Kutta Methods is constructed and some results concerning their maximum attainable order and stability properties are given.
Abstract: An efficient way of implementing Implicit Runge-Kutta Methods was proposed by Butcher [3]. He showed that the most efficient methods when using this implementation are those whose characteristic polynomial of the Runge-Kutta matrix has a single reals-fold zero. In this paper we will construct such a family of methods and give some results concerning their maximum attainable order and stability properties. Some consideration is also given to showing how these methods can be efficiently implemented and, in particular, how local error estimates can be obtained by the use of embedding techniques.
TL;DR: In this article, three types of stability of real matrices are compared and necessary conditions are obtained in terms of the principal submatrices of a real matrix for normal matrices and matrices whose off-diagonal elements are all positive.
TL;DR: In this article, it was shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x 0, x 1,…), where xk = x 0 Rk, for k ≧ 0.
Abstract: It is shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x 0, x 1,…), where xk = x 0 Rk , for k ≧ 0. The rate matrix R is an irreducible, non-negative matrix of spectral radius less than one. The matrix R is the minimal solution, in the set of non-negative matrices of spectral radius at most one, of a non-linear matrix equation. Applications to queueing theory are discussed. Detailed explicit and computationally tractable solutions for the GI/PH/1 and the SM/M/1 queue are obtained.
TL;DR: It is shown that for machines with two-dimens iona l rec tangula r grid connect ivi ty (such as I L L I A C IV), mui t tphcat lon and inversion o f NxN matrices inherent ly requi re O(N) steps, even if the process ing e lements are no t cons t ra ined to execute identical instruct ions.
Abstract: In this pape r it is shown how da ta movemen t , r a the r than ar i thmet ic opera t ions , can be the hml tmg fac tor in the pe r fo rmance o f paral lel compu te r s on matr ix computa t ions In par t icular it is p roved that for machines with two-dimens iona l rec tangula r grid connect ivi ty (such as I L L I A C IV), mui t tphcat lon and inversion o f NxN matrices inherent ly requi re O(N) steps, even if the process ing e lements are no t cons t ra ined to execute identical instruct ions.
TL;DR: In this paper, various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix, are studied.
Abstract: We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.
TL;DR: In this article, the exact matrix for the supersymmetric nonlinear nonsymmetric nonsmooth ϵ model was constructed in one space and one time dimension, and the results confirm that this model possesses mass generation and chiral symmetry breaking.
Abstract: We construct the exact $S$ matrix for the supersymmetric nonlinear $\ensuremath{\sigma}$ model in one space and one time dimension. The results confirm that this model possesses mass generation and chiral-symmetry breaking. As a byproduct, we also construct the $S$ matrix for the elementary boson and fermion of the supersymmetric form of the sine-Gordon equation.
TL;DR: In this article, a new approximation for the nuclear density matrix based on the Density Matrix Expansion (DME) of Negele and Vautherin was proposed, which gives better results than the Slater and the truncated DME approximations.
16 Oct 1978
TL;DR: A new technique of trilinear operations of aggregating, uniting and canceling is introduced and applied to constructing fast linear non-commutative algorithms for matrix multiplication and the result is an asymptotic improvement of Strassen's famous algorithms.
Abstract: A new technique of trilinear operations of aggregating, uniting and canceling is introduced and applied to constructing fast linear non-commutative algorithms for matrix multiplication. The result is an asymptotic improvement of Strassen's famous algorithms for matrix operations.
TL;DR: The exact noncentral distributions of matrix variates and latent roots derived from normal samples involve hypergeometric functions of matrix argument as discussed by the authors, which can be defined as power series, by integral representations, or as solutions of differential equations, and there is no doubt that these mathematical characterizations have been a unifying influence in multivariate noncentral distribution theory.
Abstract: The exact noncentral distributions of matrix variates and latent roots derived from normal samples involve hypergeometric functions of matrix argument. These functions can be defined as power series, by integral representations, or as solutions of differential equations, and there is no doubt that these mathematical characterizations have been a unifying influence in multivariate noncentral distribution theory, at least from an analytic point of view. From a computational and inference point of view, however, the hypergeometric functions are themselves of very limited value due primarily to the many difficulties involved in evaluating them numerically and consequently in studying the effects of population parameters on the distributions. Asymptotic results for large sample sizes or large population latent roots have so far proved to be much more useful for such problems. The purpose of this paper is to review some of the recent results obtained in these areas.