Showing papers on "Square matrix published in 1981"
TL;DR: A survey of computational methods in linear algebra can be found in this article, where the authors discuss the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, and more traditional questions such as algebraic eigenvalue problems and systems with a square matrix.
Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).
667 citations
TL;DR: In this paper, the number of independent parameters of the Mueller matrix of a collimator-radiometer system has been investigated in terms of 9 nonlinear (bilinear) equations.
191 citations
TL;DR: This algorithm is easily seen to be a generalization of the well-known QR algorithm and how it is related to the power method and inverse iteration, and for special matrices the connection between the LR and HR algorithms is indicated.
92 citations
TL;DR: In this article, the permanental polynomial of the adjacency matrix of a graph has been studied, as well as the permanents of the Laplacian matrix of the graph.
89 citations
TL;DR: It is shown that this method yields very satisfactory analyses for journal and national citation data, enabling the members of the set to be assigned measures of size, quality and self-interest and a fuzzy set of clustered members from which all data may be derived.
Abstract: A method is explained for analysing square matrices of statistics giving transactions between each member of a set of nations, papers, journals, etc. In general self-transactions are different in kind to other exchanges of money, citations, etc., and a special method is given to compute row and column coefficients without relying on the diagonal elements. It is shown that this method yields very satisfactory analyses for journal and national citation data, enabling the members of the set to be assigned measures of size, quality and self-interest and a fuzzy set of clustered members from which all data may be derived.
85 citations
TL;DR: In this article, an efficient numerical solution of the matrix equation $AX^2 + BX + C = 0, where A, B, C and X are all square matrices, is presented.
Abstract: This paper is concerned with the efficient numerical solution of the matrix equation $AX^2 + BX + C = 0$, where A, B, C and X are all square matrices. Such a matrix X is called a solvent. This equation is very closely related to the problem of finding scalars $\lambda $ and nonzero vectors x such that $(\lambda ^2 A + \lambda B + C)x = 0$. The latter equation represents a quadratic eigenvalue problem, with each $\lambda $ and x called an eigenvalue and eigenvector, respectively. Such equations have many important physical applications.By presenting an algorithm to calculate solvents, we shall show how the eigenvalue problem can be solved as a byproduct. Some comparisons are made between our algorithm and other methods currently available for solving both the solvent and eigenvalue problems. We also study the effects of rounding errors on the presented algorithm, and give some numerical examples.
73 citations
52 citations
TL;DR: In this article, it was shown that basic descriptive quantities, such as the stationary vector and mean first-passage matrix, can be calculated using any one of a class of fundamental matrices.
43 citations
TL;DR: In this article, a general design formulation for a Butler matrix (B matrix) is described, where the B matrix design problem is used to determine phase shift location and value in a matrix, when the number of beam (elements of array) M = 2−N and the scattering matrix for the hybrid couplers are specified.
Abstract: A systematic general design formulation for a Butler matrix ( B matrix) is described. The B matrix design problem discussed is used to determine phase shift location and value in a matrix, when the number of beam (elements of array) M = 2^{N} and the scattering matrix for the hybrid couplers are specified. The design formulation presented is based on the fact that a B matrix design procedure and an FFT algorithm are equivalent in fundamental concepts. It is shown that the B matrix design procedure can be systematically formulated by the FFT algorithm modifications, which preserves the topological properties of the original signal flow diagram. A simple design formula has been established by this formulation.
30 citations
TL;DR: In this paper, an extended multidimensional Newton-Raphson method is proposed for the factorization of matrix polynomials, and a root-locus approach and a matrix continued fraction approach are presented to make initial guesses for rapid convergence of the Newton Raphson Method.
Abstract: An extended multidimensional Newton-Raphson method is proposed for the factorization of matrix polynomials. A root-locus approach and a matrix continued fraction approach are presented to make initial guesses for rapid convergence of the Newton-Raphson method. The computer-aided method can be used to determine the spectral factors of a matrix polynomial for the analysis and synthesis of kinematic and dynamic systems, and to obtain the spectral factorization of a matrix polynomial for optimal control and filtering problems. The same approach can be applied to determine the nth root of a real or complex matrix.
21 citations
TL;DR: Effectes of several commonly used transformations (including percentage formation, row and column normalization, and the square root transformation) have been examined with emphasis placed on changes in the statistical and geometrical properties of column vectors that accompany the application of the transformation.
Abstract: Proper analysis of transformed data arrays (such as percentages) requires paying special attention to the effects of the transformation process itself. Effects of several commonly used transformations (including percentage formation, row and column normalization, and the square root transformation) have been examined with emphasis placed on changes in the statistical and geometrical properties of column vectors that accompany the application of the transformation. Even though many transformations, including taking the square root, “open up” the percentage array, this does not allow one to ignore the fact that percentage formation may have considerably modified the statistical and geometrical properties of the columns of the matrix. In preparing to analyze percentages one should give serious consideration to using the row normalized form of the data matrix. The individual elements in such a matrix are the direction cosines of the vector in M-dimensional space, the row vectors are of unit length, and the row normalized matrix computed from the closed array is equal to the row normalized, open matrix that is unobservable. Application of a column transformation (such as range restriction and proportion of the maximum) destroys the equality of the open and percentage row normalized matrices. Despite repeated claims to the contrary, one can not deduce the statistical and geometrical properties of the open matrix given only the statistical and geometrical properties of the closed matrix.
TL;DR: The relationship between solvents and spectral factors of a high-degree matrix polynomial is explored in this article, where various new transformations are developed to convert right (left) solvent into spectral factors and vice versa.
Abstract: The relationships between solvents and spectral factors of a high-degree matrix polynomial are explored. Various new transformations are developed to convert right (left) solvents into spectral factors and vice versa. The transformation of a right (left) solvent to a left (right) solvent is also established. The newly established algorithms are then applied to determine the spectral factorization of a matrix polynomial for optimal control problems. The developed algebraic theory enhances the capability of the analysis and synthesis of a system described by a high-degree matrix differential equation.
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01 Jan 1981
TL;DR: In this paper, a generalization of the P-matrices is proposed, where the class U is defined 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.
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)
TL;DR: In this paper, a generalized Newton method based on the contracted gradient of a matrix polynomial is derived for solving the right (left) solvents and spectral factors of matrix poynomials.
Abstract: A generalized Newton method, based on the contracted gradient of a matrix polynomial, is derived for solving the right (left) solvents and spectral factors of matrix polynomials. Two methods of selecting initial estimates for rapid convergence of the newly developed numerical method are proposed. Also, new algorithms for solving complete sets of the right (left) solvents and spectral factors without directly using the eigenvalues of matrix polynomials are derived. The proposed computer-aided method can be used to determine the spectral factorization of a matrix polynomial for optimal control, filtering and estimation problems.
TL;DR: In this paper, it was shown that the problem of deciding which matrices satisfy (1) is equivalent (even in this general context) to deciding which pairs of self-adjoint (hermitian) matrices are involutorily conjunctive.
TL;DR: In this paper, a finite power series expansion method is suggested for the evaluation of the transition matrix of a system of linear time-invariant differential equations, and an upper bound for the errors (due to round-off and truncation) involved in the computation is given.
Abstract: A finite power series expansion method is suggested for the evaluation of the transition matrix of a system of linear time-invariant differential equations. An upper bound for the errors (due to round-off and truncation) involved in the computation is devolopod. Several examples are given to illustrate the effectiveness of this approach.
TL;DR: In this paper, the determinant of the solution to the Lyapunov matrix equation was shown to be the same as in [5] and [6], and they gave another bound for this value.
Abstract: In [5] bound for the determinant of the solution to the Lyapunov matrix equation was reported. This note gives an another bound for this value.
TL;DR: In this article, it is shown that by dynamic feedback the closed loop transfer matrix of a linear system can be made equal to a proper rational matrix of the form v(sT'−1(s)T−(s),T k (s)
Abstract: It is shown that by dynamic feedback the closed loop transfer matrix of a linear system can be made equal to a proper rational matrix of the form v(s)T'−1(s)T−(s)T k (s) Here V(s) is the numerator polynomial matrix of the open loop transfer matrix, T'(s) is a polynomial matrix which can be chosen arbitrarily up to some degree constraints, T'2(s) is a polynomial matrix whose invariant factors can be chosen arbitrarily up to some degree inequalities, T k(s) is a polynomial matrix which is the denominator matrix in a matrix fraction description of the feedback matrix
01 Jan 1981
TL;DR: In this paper, the authors present comments on qualitative stability of matrices and economic theory and show that qualitative stability is associated with the special qualitative structure of the matrices, which are associated with some phase of economic theory.
Abstract: Publisher Summary This chapter presents comments on qualitative stability of matrices and economic theory. Various kinds of stability are always associated with the special qualitative structure of the matrices, which are associated with some phase of economic theory. The matrices under consideration are sometimes dense and have special structures. They are square matrices and the kinds of transformations applied to such matrices are iterative multiplications and additions or subtractions. Two situations can arise if the computer is to be used to assist in model analysis: (1) verification that the structure and signs appear as initially prescribed and (2) verification that the signs remain as prescribed after some computation.
TL;DR: In this article, the authors studied the relation between the similarity invariants of a square matrix and the congruence invariant of its symmetric factors and obtained bounds for the elementary divisor structure of A in terms of the index or signature of one or both of its symmetry factors.
TL;DR: A new algorithm has been developed for triangularisation of the banded matrix that is computationally economical and advantageous as compared to the existing known algorithms.
TL;DR: In this article, a method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement.
Abstract: The problem of pole assignment in a completely controllable linear time-invariant system dx/t = Ax + Bu, y = Cx is considered. A method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement. The method does not require solution of sets of non-linear equations or manipulation of polynomial matrices, and no knowledge of eigenvalues and/or eigenvectors of A is necessary. If the computed matrix K and the given matrix C satisfy a consistency condition, a matrix Kν such that KνC = K can be directly obtained from K and the desired pole-placement can be realized by an output feedback law u = r + Kνy.
TL;DR: In this article, the authors survey a portion of the recent literature on (υ, k, λ)-designs and discuss such varied topics as integral solutions, completion theorems and λ-designs.
TL;DR: In this article, the authors deal with a system of ordinary differential equations with known initial conditions associated with a given square matrix, by using standard analytical and computational methods many of the important aspects of the given matrix can be determined.
01 Jan 1981
TL;DR: The Lancaster original standard form (LOSF) is a triangular square matrix with positive signs above the diagonal and a negative diagonal, augmented by a final column of positive entries as discussed by the authors.
Abstract: Publisher Summary This chapter discusses Maybee's sign solvability. The Lancaster Original Standard Form (LOSF) form is a triangular square matrix with positive signs above the diagonal and a negative diagonal, augmented by a final column of positive entries. All square matrices, for which all paths are positive and in which there are no positive cycles of dimension greater than one can be put in form M2 by permissible permutations and sign changes. The original LOSF is both necessary and sufficient for sign solvability if h ≪ 0, but the general Lancaster general standard form is not the most general form as it is sufficient, but not necessary, for sign solvability in cases where there are no columns without zeros.
01 Aug 1981
TL;DR: In this article, a method for inverting any polynomial matrix, including cases in which the lowest and highest matrix coefficients are both singular, is described, and an application to network sensitivity analysis is presented.
Abstract: A method for inverting any polynomial matrix, including cases in which the lowest and highest matrix coefficients are both singular, is described. An application to network sensitivity analysis is presented.
TL;DR: In this paper, necessary and sufficient conditions under which a given square matrix A(t) can be decomposed into the product, exp (A1t)A(0) exp ( ) where Ax is a constant matrix are investigated.
Abstract: Necessary and sufficient conditions under which a given square matrix A(t) can be decomposed into the product, exp (A1t)A(0) exp ( — A1l) where Ax is a constant matrix are investigated. The state transition matrix of the state equations [Xdot](t) = A(t)X(t) can be expressed as φ(t.t0) = exp (A1l) exp A2(t —t0) exp (— A1l) with A2δA(0) — A1if and only if A(t) has such a decomposition. In this paper some properties of the system described by [Xdot](t) = A(t)X(t) are examined and necessary and sufficient conditions for A(i) to be decomposable are given; an algorithm for determining A1is also provided
TL;DR: In this article, it was shown that for any n-diagonal matrix A, over a field F of characteristic + 2, the matrix DA has a multiple eigenvalue (or a multiple permanental root).