Showing papers on "Matrix analysis published in 1983"
01 Dec 1983
TL;DR: In this paper, a simple algorithm for computing generalized inverses of a constant matrix is established, and then applied to the case of matrices having polynomial elements in several variables.
Abstract: : Theory and computation techniques of the various type of generalized inverses of matrices which have polynomial elements x, y, z ..., etc., are presented. A simple algorithm for computation of generalized inverses of a constant matrix is established, and then applied to the case of matrices having polynomial elements in several variables. Reduction of a matrix to its smith form over the ring of polynomial elements in several variables is presented. A simple algorithm for investigation of the system Ax = b in case of constant and nonconstand rank of A is presented. Application of generalized inverses to solve more general matrix equations such as Lyapunov and Riccati equations is studied. (Author)
113 citations
TL;DR: In this paper, the sign matrices uniquely associated with the matrices (M − ζ j I ) 2, where the corners of a rectangle oriented at π /4 to the axes of a Cartesian coordinate system, were used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigen values.
44 citations
01 Sep 1983
TL;DR: In this paper, the usefulness of matrix transformation has been noted in the design of digital filters from analog prototypes, and a general property that relates two such matrices of successive orders, (n - 1) and n, is proved.
Abstract: The usefulness of a matrix transformation has been noted in the design of digital filters from analog prototypes. In this letter, a new general property, that relates two such matrices of successive orders, (n - 1) and n, is proved.
35 citations
01 Jan 1983
32 citations
TL;DR: In this article, a new approach to the representation of nonsymmetrical optical systems by matrices is introduced, where each component of an optical system is represented by a 4 × 4 unitary matrix, and the product of those matrices yields the transfer matrix of the system.
Abstract: A new approach to the representation of nonsymmetrical optical systems by matrices is introduced. In the paraxial approximation each component of an optical system is represented by a 4 × 4 unitary matrix, and the product of those matrices yields the transfer matrix of the system. The transfer matrix that represents the propagation between two arbitrary planes through the system containing two independently rotated cylindrical lenses is decomposed into the product of three matrices. The eigenvalues of the submatrices in this factorized form determine the focal lengths of the equivalent system and the localization of the foci of the system with respect to these arbitrarily chosen planes.
31 citations
TL;DR: In this article, the authors describe the properties of the class of n × n matrices such that some or all of the signs of the nonzero elements of the inverse matrix can be determined based only upon the knowledge of the sign of the matrix being inverted.
19 citations
TL;DR: In this paper, the problem of classifying pairs of nxn complex matrices (A,B) under a simultaneous similarity was solved completely and explicitly, and the classification decomposes to a finite number of steps.
Abstract: : In this paper we solve completely and explicitly the long standing problem of classifying pairs of nxn complex matrices (A,B) under a simultaneous similarity. Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set. Then we construct a finite number of rational functions in the entries of A and B whose values are constant on all pairs similar to (A,B). The values of the functions phi sub i (A,B), i equals 1,...,s, determine a finite number of similarity classes.
19 citations
TL;DR: In this paper, the authors present a selective survey of both vectors and matrices in the statistical component of econometrics, focusing on the importance of matrix algebra in the economic theory underlying econometric relations.
Abstract: Publisher Summary Vectors and matrices played a minor role in the econometric literature published before Second World War, but they have become an indispensable tool in the past several decades. Part of this development results from the importance of matrix tools for the statistical component of econometrics; another reason is the increased use of matrix algebra in the economic theory underlying econometric relations. This chapter presents a selective survey of both areas. It reviews the concepts of linear dependence and orthogonality of vectors and the rank of a matrix. A major reason related to the usefulness of matrix methods is that many topics in econometrics have a multivariate character. The chapter illustrates the convenience of matrices for linear systems. The expression “linear algebra” should not be interpreted in the sense that matrices are useful for linear systems only. Vectors and matrices are important in the statistical component of econometrics. A general method of estimation is maximum likelihood (ML) that can be shown to have certain optimal properties for large samples under relatively weak conditions. The derivation of the ML estimates and their large sample covariance matrix involves the information matrix, which is (apart from sign) the expectation of the matrix of second-order derivatives of the log-likelihood function with respect to the parameters. The prominence of ML estimation in recent years has greatly contributed to the increased use of matrix methods in econometrics.
18 citations
TL;DR: The Gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread as discussed by the authors, and it is shown that for real symmetric matrices (or complex Hermitian matrices) the ratio between the bound and the spread is bounded by p+1, where p is the maximum number of off diagonal nonzeros in any row of a matrix.
14 citations
TL;DR: In this paper, the authors extended the theory of graphs associated with real rectangular matrices to include information about the signs of the elements and showed when signed row and column graphs can be defined for the matrix A and deduced conditions under which these graphs are balanced.
Abstract: This paper extends the theory of graphs associated with real rectangular matrices to include information about the signs of the elements. We show when signed row and column graphs can be defined for the matrix A. We also deduce conditions under which these graphs are balanced. This leads to a definition of the class of quasi-Morishima rectangular matrices A. It is shown that the Perron–Frobenius theorem applies to the matrices $AA^T $ and $A^T A$ when A is a quasi-Morishima matrix. Finally we examine the applications of our results to several classes of matrices occurring in energy economic models. All results in this paper are purely qualitative in character.
TL;DR: A fast algorithm, to achieve one step of Newton's method, is shown to be suitable to compute the eigenvalues of a (2k+1)-diagonal Toeplitz matrix with elements in the complex field.
Abstract: We exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a (2k+1)-diagonal Toeplitz matrix with elements in the complex field, either for sequential or for parallel computations. A fast algorithm, to achieve one step of Newton's method, is shown to be suitable to compute the eigenvalues of such a matrix. Bounds to the eigenvalues and necessary and sufficient conditions for positive definiteness, which are easy to check, are given either for matrices with scalar elements or for matrices with blocks. In the case in which the blocks are themselves band Toeplitz matrices such conditions assume a very simple form.
22 Jun 1983
TL;DR: In this article, some transformations of matrix equivalence arising from problems in linear systems theory are discussed, and applications of the results to the linear systems area are made in the context of generalised state space systems and connections with previous notions of equivalence are developed.
Abstract: Some transformations of matrix equivalence arising from problems in linear systems theory are discussed. Such transformations not considered in the standard matrix literature relate the structure of matrices of different dimensions. Application of the results to the linear systems area is made in the context of generalised state space systems and the connection with previous notions of equivalence is developed.
TL;DR: In this article, the effectiveness of two coefficient-of-ergodiciiy bounds for the non-unit eigenvalues of a doubly stochastic matrix was compared.
Abstract: In this note we compare the effectiveness of two coefficient-of-ergodiciiy bounds for the non-unit eigenvalues of a doubly stochastic matrix, and consider an extension to ordinary stochastic matrices.
TL;DR: In this article, a definition for the characteristic equation of an N-partitioned matrix is given, and it is proved that this matrix satisfies its own characteristic equation, which can then be regarded as a version of the Cayley-Hamilton theorem, of use with N -dimensional systems.
Abstract: A definition is given for the characteristic equation of an N -partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use with N -dimensional systems.
TL;DR: A connection between the additive complexity of a system with coefficient matrix A and of aSystem of linear forms with coefficient matrices AT is proved.
Abstract: The additive computation of a system of linear forms can be represented by a sequence of square matrices Q1,...,QT (Qi equals the identity matrix increased or decreased by 1 in one entry). The complexity of the additive computation is the minimal number of matrices in such a representation. A connection between the additive complexity of a system with coefficient matrix A and of a system with coefficient matrix AT is proved.
TL;DR: In this article, conditions under which a given real square matrix A cannot be represented in the form A=D0TU, where D, 0, T are diagonal, orthogonal, and triangular real matrices, respectively, while U is a unimodular integral matrix.
Abstract: One finds conditions under which a given real square matrix A cannot be represented in the form A=D0TU, where D, 0, T are diagonal, orthogonal, and triangular real matrices, respectively, while U is a unimodular integral matrix. From these one derives the statement formulated in the title.
01 Sep 1983
TL;DR: In this article, the 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX, and transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first-order system.
Abstract: In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.
TL;DR: In this article, the generalized inverse of the product of two integral EPr matrices is shown to be integral and EPr, and necessary and sufficient conditions for it to be EPr.
Abstract: This paper gives a characterization of integralEPr matrices and necessary and sufficient conditions for the generalized inverse of the product of two integralEPr matrices to be integral andEPr.
01 Jan 1983
TL;DR: In this paper, the authors have indicated several areas in solid state physics where random matrices appear, such as spin wave propagation in disordered Heisenberg magnets and random matrix diagonalization of sparse matrices.
Abstract: We have indicated several areas in solid state physics where random matrices appear. Another closely related area not touched above is that of spin waves in disordered Heisenberg magnets [19]. Analytical and numerical techniques used in the problems mentioned above and the technique of diagonalization of sparse matrices are often useful for such problems.
TL;DR: The eigenvalues of a wave-digital filter (WDF) can be computed from two different forms of system matrices which are related by similarity transformations.
Abstract: The eigenvalues of a wave-digital filter (WDF) can be computed from two different forms of system matrices which are related by similarity transformations.
TL;DR: In this article, a Hill's matrix Lϱp of odd order corresponding to the theory of Hill's equations is considered and necessary and sufficient conditions on the coefficients of such a matrix are established in order that the matrices L1 and L−1 have all double eigenvalues except one.