Showing papers on "Square matrix published in 1973"

An Algorithm for Generalized Matrix Eigenvalue Problems.

Abstract: A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $\lambda $ are also mentioned.

Minimization of Multidimensional Linear Iterative Circuits

Abstract: A model for two-dimensional linear iterative circuits is defined in the form of matrix equations. From the matrix equations, a two-dimensional characteristic function is defined. It is then proved that a matrix satisfies its two-dimensional characteristic function. This property is used to form a diagnostic matrix. Finally, the diagnostic matrix is used in a minimization technique.

Some classes of matrices in linear complementarity theory

Abstract: The linear complementarity problem is the problem of finding solutionsw, z tow = q + Mz, w≥0,z≥0, andw T z=0, whereq is ann-dimensional constant column, andM is a given square matrix of dimensionn. In this paper, the author introduces a class of matrices such that for anyM in this class a solution to the above problem exists for all feasibleq, and such that Lemke's algorithm will yield a solution or demonstrate infeasibility. This class is a refinement of that introduced and characterized by Eaves. It is also shown that for someM in this class, there is an even number of solutions for all nondegenerateq, and that matrices for general quadratic programs and matrices for polymatrix games nicely relate to these matrices.

Representation of Concurrency with Ordering Matrices

Abstract: A formal technique for the representation of tasks such that the potential concurrency of the task is detectable, and hence exploitable, during the execution of the task is described. Instructions are represented as a pair of binary vectors d? and e which completely describe the sources and sinks specified by the instruction. Tasks are represented as square matrices M called ordering matrices. The values of the elements of these matrices are used to dynamically indicate the necessary ordering of the execution of instructions. It is shown how several different types of ordering matrices, each type having the capability of exhibiting different amounts of potential concurrency, can be calculated from the d? and e vectors of the instructions of a task using ``linear algebraic-like'' operations. For example, intercycle independencies can be detected with a ternary ordering matrix. This matrix can be extended to dynamically detect opportunities for reassigning the resources specified by certain instructions to increase the amount of potential concurrency. Experimental results are presented showing the relative capability of each of these matrix types for exhibiting potential concurrency. These techniques are shown to produce somewhat greater amounts of potential concurrency than other known dynamic techniques. However, the amounts of potential concurrency found are less than those reported for preprocessing detection techniques.

Optimal design of geodetic nets, 2

Abstract: Let the configuration or first-order design matrix A of a geodetic net be given and let the weight or second-order design matrix P of m observations be unknown. For some predetermined choice of the dispersion matrix Σ of the n unknown net coordinates, the unique minimum Euclidean norm solution for the weight matrix P is P = (A′)+Σ−1A+. A′ indicates the transpose and A+ the pseudoinverse of the (in general) rectangular matrix A. For a general first-order design matrix A and a given Σ, there does not exist a solution for a diagonal positive definite weight matrix P. Our solution for P belongs to the first category of an optimal design of C. R. Rao. The designs P = I and P = (A′)+Σ−1A+ yield the same best linear unbiased estimator for the unknowns, but the variance covariance matrix for these designs is different. Examples, especially the structure by G. I. Taylor and T. Karman for the homogeneous and isotropic geodetic net, illustrate theoretical results.

Analytic solution of spatially discretized groundwater flow equations

Abstract: The Galerkin procedure when it is applied to the equation for horizontal two-dimensional flow of groundwater in a nonhomogeneous isotropic aquifer generates approximating equations of the following form: Rc + G [dc/dt] + f = 0, where R and G are square matrices, c and f are column matrices, and t is time. This matrix equation is decoupled and solved for the unknown column matrix c(t). In the case of a confined aquifer that approaches a steady state solution, R, G, and f are constant. An analytic solution to the matrix equation for c(t) is given for this case. In the case of a water table aquifer that approaches a steady state solution, R and f are explicitly dependent on c(t), and G is constant. For this case, c(t = ∞) is found in a simple iterative manner, and an iterative procedure is given to approximate c(t). These methods are compared with the approximate numerical Crank-Nicholson procedure by applying both to a particular problem for which the unknown column matrix c(t) has 49 elements. The Crank-Nicholson procedure is found usually to require less computation time to evaluate c(t) for the confined aquifer case but to give errors for drawdown averaging approximately 10%. The Crank-Nicholson procedure is found to take considerably more computation time to evaluate c(t =∞) for both the confined and the water table cases but to take considerably less time to evaluate c(t) for the water table case.

Another Proof of the Two-Dimensional Cayley-Hamilton Theorem

Abstract: Givone and Roesser [1] state a "two-dimensional" analog of the Cayley-Hamilton theorem: every square matrix satisfies its characteristic equation. Readers may be interested in a proof different from the one given in [1].

Estimation of standard errors of the characteristic roots of a dynamic econometric model

Abstract: where y(t) represents the vector of endogenous variables, x(t) the vector of exogenous variables, u(t) the vector of stochastic disturbances, and t the tth period of observation. The matrices A, (T = 0, 1, . . . , m) of the structural coefficients are square matrices of order G. It is assumed that the conditions justifying the theorems in [3, Ch. 10] are satisfied, and that there are no nonlinear restrictions on the elements of A.. The stability of the system is determined by reference to the dominant root of the polynomial equation (2) det E Atmt) =0. t=O

A new method of parameter estimation in linear systems

Abstract: A time domain method is given for estimating the matrix of related parameters in linear systems with constant coefficients and real eigenvalues. The method consists of a one-dimensional search for the local minima of a scalar function μ(λ), which provide the eigenvalues of the system matrix and the matrix itself when observable. Applications are given to the determination of a transfer function and the estimation of the rate matrix of a monomolecular reaction system. Questions of accuracy, number, and type of measurements required are discussed.

Study of methods of computing transition matrices

Abstract: Various methods of computing the square matrices ? = eAT and ? = ?T0 e A? d? are studied. They are programmed in FORTRAN and compared for storage, speed and accuracy on a variety of test matrices. The methods studied are power-series expansion and related approximants; the eigenvalue methods of Sylvester's expansion and diagonal transformation; and numerical integration. The rational approximant to the power-series expansion and a block-diagonal transformation are shown to be the superior methods.

Comments on "A Fast Computer Method for Matrix Transposing" and Application to the Solution of Poisson's Equation

Abstract: In the above correspondence,1an algorithm has been described that allows the transposing of a 2n× 2nmatrix by reading and writing n times at the most via direct access rows of the matrix and by permuting its data elements. The generalization to arbitrary square matrices has not been described, and the generalization to nonsquare matrices has been reported to be impossible.

The Jacobi-Perron algorithm and the algebra of recursive sequences

Abstract: This paper shows the existence of a one-to-one correspondence between a certain class of square matrices of arbitrary order and a related extension field. The elements of these matrices are obtained from certain basic linear recursive sequences by means of a generalization of the euclidean algorithm.

On monotone matrix functions of two variables

Abstract: The theory of monotone matrix functions has been developed by K. Loewner; he first gives some necessary and sufficient conditions for a function to be a monotone matrix function of order n, and then, as a result of further deep investigations including questions of interpolation he arrives at the following criterion: A real-valued function f(x) defined in (a, b) is monotone of arbitrary high order n if and only if it is analytic in (a, b), can be analytically continued onto the entire upper half-plane, and has there a nonnegative imaginary part. The problem of monotone operator functions of two real variables has recently been considered by A. Koranyi. He has generalized Loewner's theorem on monotone matrix functions of arbitrary high order n to two variables. We seek a theory of monotone matrix functions of two variables analogous to that developed by Loewner and show that a complete analogue to Loewner's theory exists in two dimensions.

On real matrices with positive definite symmetric component

Abstract: Inequalities concerning real square matrices A with positive definite symmetric component A+A ∗are derived from certain inertia relations which hold for any complex (not necessarily real) square matrices A with positive definite A+A ∗

A Relation Between the Moore–Penrose and Commuting Reciprocal Inverses

Abstract: If a singular square matrix has both a commuting reciprocal and Moore–Penrose inverse, it is proved that either of these may be factorized in terms of the other. Examples related to singular boundary value problems are given.

Propagation Analysis of HF Currents and Voltages on Lossy Power Lines

Abstract: Matrix analysis is described of voltage and current propagations along a resistive conductor system above an imperfectly conducting ground. Square matrices and their functions are considered as operators in a vector space. Conductor voltages and currents are represented as vectors and resolved into modes or eigenvectors of the propagation matrix. It is shown that although the modal transformation is hot power invariant the mode are independent but not orthogonal. General and modal reflection-free line terminations are developed. A brief review of some simplified modal analyses is described. A numerical solution is demonstrated. Mathematical background is appended.

A Note on Pairs of Matrices and Matrices of Monotone Kind

Abstract: Several new results are presented concerning matrices having nonnegative inverses. We consider the problem of determining conditions on the pair of real square matrices $(A,B)$ such that for every diagonal matrix D with positive elements on the main diagonal, the inverse of the matrix $AD + B$ has only nonnegative elements. Of primary concern are the two special cases which occur frequently in the applications, (a) the case in which A is the identity matrix, and (b) the case in which both of the matrices $A,B$ have only nonpositive off-diagonal elements.

Note on a paper by V.S. Godlevskii

Abstract: In what follows the word matrix denotes a square matrix of order n with complex elements, and the word vector denotes an n-dimensional vector with complex components. The norm of a vector and its subordinate operator norm of matrices, are fixed; this operator norm of matrices is associated, as in [21, with the logarithmic norm. The symbols A, B, 2 denote matrices; x, y, 4, 7 denote vectors. The symbols \]A\\, [lx\\ and y(A) denote, respectively, the norm of A, the norm of x and the logarithmic norm of A.

Digital simulation of the rotation of a moving triad with a known matrix spin vector

Abstract: This paper discusses the numerical evaluation of the elements of the transformation matrix of an orthogonal rectilinear triad moving with respect to a similar fixed system, when the former rotates with a known spin vector Typical governing equations are formu lated in terms of Euler angles, in terms of quater nions, and in terms of direction cosines and their derivatives Emphasis is placed on the latter formu lation for solution by digital computationThe governing equations are reformulated into a homogeneous and linear matricial state variable equa tion of the first order, with a skew symmetric coefficient matrix called the rotation operator The transition matrix of this state equation gives the required rotation matrix The characteristics of the skew-symmetric matrices are used in the development of a suitable algorithm for the solutionThe exact rotation matrix can be obtained for cases of constant spin vector and for special cases of variable spin vectors For the general case of a variable spin

Identification of the return difference matrix of a multivariable linear control system, in terms of its inverse closed‐loop transfer matrix

Abstract: The return difference matrix F(s) of a multivariable control system with respect to its gain elements is obtainable directly from the inverse transfer matrix T−1(s), which is in turn derived (by elimination of undriven node groups) from the coefficient matrix of the set of independent differential equations describing the system. Thus, no knowledge of the system's topology is required in order to obtain F(s). Furthermore, the inverse of one of the submatrices of T−1(s) yields the idealized response, if all gain elements are assumed to exhibit infinite gain.

Digital filter synthesis--A low-sensitivity system matrix

Abstract: A new system matrix that can realize any second-order digital filter is presented. This matrix exhibits minimum eigenvalue sensitivity when compared to other second-order matrices over certain regions of root locations within the unit circle.

On the utility of separable expansions for the t matrix

Abstract: Osborn used the property of compactness to show that there is an infinite mean square deviation between the t matrix for a local potential and any separable t matrix of finite rank. I present an alternate proof, using only the property of square integrability. The divergence of the mean square deviation arises from very large momenta. I argue that a separable t matrix can give a good approximation to the trinucleon energy, which is insensitive to values of the t matrix at very large momenta.

Generalized Method for Calculating Löwdin Orbitals

Abstract: A generalized method for numerically calculating Lowdin orbitals is proposed, the method being an application of Frobenius’ theorem in algebra. Lowdin transformation matrix T=S−1⁄2 (S is overlap matrix) is calculated as T=US0−1⁄2 Ut, where S0 is a diagonal matrix whose diagonal elements are eigen values of S, U is a matrix whose column vectors are eigen vectors of S and Ut is the transposed matrix of U. It is proved that the eigen values of the overlap matrix of S are always positive for any choice of linearly independent atomic orbitals and that S−1⁄2, the inverse square-root of S, exists within the range of real numbers. This method is useful for computer calculation by applying Jacobi’s method.