Showing papers on "Matrix analysis published in 1981"

Row Stochastic Matrices Similar to Doubly Stochastic Matrices

Abstract: The problem of determining which row stochastic n-by-n matrices are similar to doubly stochastic matrices is considered. That not all are is indicated by example, and an abstract characterization as well as various explicit sufficient conditions are given. For example, if a row stochastic matrix has no entry smaller than (n+1)−1 it is similar to a doubly stochastic matrix. Relaxing the nonnegativity requirement, the real matrices which are similar to real matrices with row and column sums one are then characterized, and it is observed that all row stochastic matrices have this property. Some remarks are then made on the nonnegative eigenvalue problem with respect to i) a necessary trace inequality and ii) removing zeroes from the spectrum.

On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings

Abstract: Various types of transfer matrix factorizations are of interest when designing regulators for generalized types of linear systems (delay differential, 2-D, and families of systems). This paper studies the existence of stable and of stable proper factorizations, in the context of the theory of systems over rings. Factorability is related to stabilizability and detectability properties of realizations of the transfer matrix.

A Simultaneous Iteration Algorithm for Real Matrices

Abstract: Simultaneous iteration methods are extensions of the power method whereby iteration is carried out with a number of trial vectors tha t converge onto the eigenvectors corresponding to the dominant eigenvalues. T h e y are particularly suitable where the matrix requiring eigensolution is large and sparse, or where approximations to the required dominant eigenvectors are already available. The first s imultaneous iteration methods are due to Bauer [1] and have been discussed by Wilkinson [10]. Subsequent methods are related to Bauer ' s bi-iteration technique but improve the convergence rate by including an interaction analysis into the iteration cycle. These methods have been most highly developed for the real syrmnetric eigenvalue problem where the left and right eigenvectors coincide. A previous paper by the authors [4] describes two possible pro~cedures for real unsymmetr ic eigenvalue problems. One procedure is a bi-iteration technique in which left and right eigenvector sets are simultaneously predicted, while the

Computer-Oriented Formulation of Transition-Rate Matrices via Kronecker Algebra

Abstract: This paper formulates the differential equations typical of a Markov problem in system-reliability theory in a systematic way in order to generate computer-oriented procedures. The coefficient matrix of these equations (the transition rate matrix) can be obtained for the whole system through algebraic operations on component transition-rate matrices. Such algebraic operations are performed according to the rules of Kronecker Algebra. We consider system reliability and availability with stress dependence and maintenance policies. Theorems are given for constructing the system matrix in four cases: * Reliability and availability with on-line multiple or single maintenance. * Reliability and availability with system-state dependent failure rates. * Reliabilityand availability with standby components. * Off-line maintainability. The results are expres § ed in algebraic terms and as a consequence their implementation by a computer program is straightforward. We also obtain information about the structure of the matrices involved. Such information can considerably improve computational efficiency of the computer codes because it allows introducing special ideas and techniques developed for large-system analysis such as sparsity, decomposition, and tearing.

Transfer function matrix of 2-D systems

Abstract: A formula is given which allows the determination of the transfer function matrix of linear multivariable two-dimensional (2-D) systems directly in terms of the state transition matrix and the characteristic equation.

Scattering Matrix Analysis of Surface Acoustic Wave Reflectors and Transducers

Abstract: Absmcr-The phased periodic array scattering parameters [l] have been modified and corrected for use as a single electrode mixed units scattering matrix consisting of one electrical and two acoustic ports. These scattering parameter analytical expressions are functions of material constants, electrode geometry, and frequency. The threeport single-electrode scattering matrices are acoustically cascaded to produce a three-port device scattering mat&x which takes into account all finger interactions (acoustic and electric) for the combined effects of piezoelectric and mechanical scattering. The analysis agrees well with experimental measurements of input admittance, electroacoustic transfer function, and acoustic transmission and reflection coefficients as functions of frequency. Analysis results for the complete modeling of solid and split electrode transducers and transducers with floating electrodes are also presented.

Geometric aspects of the linear complementarity problem

Abstract: : A large part of the study of the Linear Complementarity Problem (LCP) has been concerned with matrix classes. A classic result of Samelson, Thrall, and Wesler is that the real square matrices with positive principal minors (P-matrices) are exactly those matrices M for which the LCP (q,M) has a unique solution for all real vectors q. Taking this geometrical characterization of the P-matrices and weakening, in an appropriate manner, some of the conditions, we obtain and study other useful and broad matrix classes thus enhancing our understanding of the LCP. In Chapter 2, we consider a generalization of the P-matrices by defining the class U as all real square matrices M where, if for all vectors x within some open ball around the vector q the LCP (x,M) has a solution, then (q,M) has a unique solution. We develop a characterization of U along with more specialized conditions on a matrix for sufficiency or necessity of being in U. Chapter 3 is concerned with the introduction and characterization of the class INS. The class INS is a generalization of U gotten by requiring that the appropriate LCP's (q,M) have exactly k solutions, for some positive integer k depending only on M. Hence, U is exactly those matrices belonging to INS with k equal to one. Chapter 4 continues the study of the matrices in INS. The range of values for k, the set of q where (q,M) does not have k solutions, and the multiple partitioning structure of the complementary cones associated with the problem are central topics discussed. Chapter 5 discusses these new classes in light of known LCP theory, and reviews its better known matrix classes. Chapter 6 considers some problems which remain open. (author)

Inversion of polynomial matrices

Abstract: A new method for the inversion of any polynomial matrix is given. This method requires operations on constant matrices only and is suitable for computer programming.

An easy way to find gradient matrix of composite matricial functions

Abstract: A simple method to compute the gradient matrix of scalar-valued composite function of matrices is presented. Two examples are provided to show the efficiency of the approach.

A note on the reduced Schur-Cohn criterion

Abstract: This correspondence simplifies the reduced Schur-Cohn matrices of an earlier published paper [1]. In particular, the symmetric matrices B for n -even, and the A -matrix for n -odd in connection with theorems l e and l o of [1] are simplified.

Optimization of Functions of Matrices with Applications to Statistical Problems.

Abstract: : The paper reviews some methods of optimizing functions of vectors and matrices subject to some restrictions and develops techniques for solving them without using the calculus of matrix derivatives. Their application to a number of statistical problems in linear estimation and multivariate analysis is illustrated. (Author)

The computation of a transfer function matrix from the given state equations for time-delays systems

Abstract: An algorithm is established to compute the transfer function matrix from the state equations for linear multivariable system with constant noncommensurable delays The proposed algorithm requires operations with constant matrices only and is suitable for computer programming An example illustrating the proposed algorithm is given

An algorithm for computing functions of triangular matrices

Abstract: Any function of a triangular matrix can be recursively calculated on the basis of simple properties stated in this paper.

Linear programming by an effective method using triangular matrices

Abstract: A modification of the projection method for linear programming is presented. This modification determines the step direction by solving two triangular systems of linear equations. The triangular matrix is updated in each step by deleting a row and adding a new one whose elements were already computed for the step-size determination. Thus there is no real computational effort in the matrix-updating. The size of the triangular systems depends on how many of the active constraints have become active after the constraint that’s going to become inactive. In the worst case, i.e. if the oldest active constraint becomes inactive, the computational effort in solving the triangular systems corresponds to that of the matrix-updating in the projection method, whereas in all other cases the effort is reduced. This reduction can be very high. Cycling of the method is excluded by a very simple rule.

A minimum realization method

Abstract: A new method of obtaining a minimum state space ( A, B, C, D ) realization of an r \times m proper rational transfer function matrix, P(s) , is presented. D is found in the usual manner. The denominator roots are calculated and the A matrix is formed. An initial estimate of the B and C matrices is assigned and a transfer function matrix is calculated from the estimated state space matrices. The B and C matrices are adjusted by the algorithm until the computed transfer function is "close enough" to the original transfer function matrix.

Some remarkable spin-1/2-like algebraic properties of spin-3/2 matrices

Abstract: Using for spin-3/2 matrices a direct-product structure involving the usual Pauli spin matrices, the authors derive the Dirac-Clifford matrices in terms of certain algebraic combinations of spin-3/2 matrices in a representation-independent way, thus achieving an extension of the Pauli spin matrices from the usual spin-1/2 space to the spin-3/2 space. Basing the derivation directly on this analysis, two algebras satisfied by spin-3/2 matrices are derived. One of these, which is also satisfied by spin-1/2 matrices, is directly related to the spin-3/2 algebras of Weaver (1978) and of Bhabha and Madhava Rao (for three objects). The other algebra is new and curiously is not satisfied by spin-1/2 matrices.

On relations and matrices

Abstract: The use of matrices to represent relations is discussed Connections are found between relations under relation composition and matrices under matrix multiplication

Research on numerical algorithms for large space structures

Abstract: Numerical algorithms for analysis and design of large space structures are investigated. The sign algorithm and its application to decoupling of differential equations are presented. The generalized sign algorithm is given and its application to several problems discussed. The Laplace transforms of matrix functions and the diagonalization procedure for a finite element equation are discussed. The diagonalization of matrix polynomials is considered. The quadrature method and Laplace transforms is discussed and the identification of linear systems by the quadrature method investigated.

An algorithm for computing singular values of large matrices for use in the analysis of large systems

Abstract: An algorithm for computing a few or many of the singular values (and a few of the corresponding singular vectors) of large matrices is presented. If the matrices under consideration are sparse, then this procedure has storage requirements that increase only linearly with the order of the matrix. Such an algorithm may prove useful in sensitivity and stability analyses of very large systems.