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Showing papers on "Square matrix published in 1985"


Journal ArticleDOI
TL;DR: In this paper, the conditions under which unique differentiable functions λ(X) and u(X), respectively, exist in a neighborhood of a square matrix (complex or otherwise) satisfying the equations Xu = λu, λO, and Xu = ǫ, were investigated.
Abstract: Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, . We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and . We obtain the first and second derivatives of λ(X) and the first derivative of u(X). Two alternative expressions for the first derivative of λ(X) are also presented.

224 citations


Book ChapterDOI
01 Jan 1985
TL;DR: In this article, the authors study the different forms of line-sum-symmetric scalings of square nonnegative matrices and characterize matrices for which such scalings exist, up to a scalar multiple of the blocks corresponding to the classes of the given matrix.
Abstract: A square matrix is called line-sum-symmetric if the sum of elements in each of its rows equals the sum of elements in the corresponding column. Let A be an n×n nonnegative matrix and let X and Y be n×n diagonal matrices having positive diagonal elements. Then the matrices XA, XAX−1 and XAY are called a row-scaling, a similarity-scaling and an equivalence-scaling of A. The purpose of this paper is to study the different forms of line-sum-symmetric scalings of square nonnegative matrices. In particular, we characterize matrices for which such scalings exist and show uniqueness of similarity-scalings and uniqueness of row-scalings, up to a scalar multiple of the blocks corresponding to the classes of the given matrix.

63 citations


Journal ArticleDOI
TL;DR: In this paper, necessary and sufficient conditions for closed-loop eigenstructure assignment by output feedback in time-invariant linear multivariable control systems are presented, where the concept of inner inverse of a matrix is employed to obtain a condition concerning the assignment of an eigen structure consisting of the eigenvalues and a mixture of left and right eigenvectors.
Abstract: Some necessary and sufficient conditions for closed-loop eigenstructure assignment by output feedback in time-invariant linear multivariable control systems are presented. A simple condition on a square matrix necessary and sufficient for it to be the closed-loop plant matrix of a given system with some output feedback is the basis of the paper. Some known results on entire eigenstructure assignment are deduced from this. The concept of an inner inverse of a matrix is employed to obtain a condition concerning the assignment of an eigenstructure consisting of the eigenvalues and a mixture of left and right eigenvectors.

51 citations


Journal ArticleDOI
TL;DR: A condition which is necessary and sufficient for the strong linear independence of columns of a given matrix in the minimax algebra based on a dense linearly ordered commutative group and a connection with the classical assignment problem is formulated.

45 citations


Journal ArticleDOI
TL;DR: The question of whether a real matrix is symmetrizable via multiplication by a diagonal matrix with positive diagonal entries was reduced to the corresponding question for M -matrices and related to Hadamard products in this paper.

45 citations



Journal ArticleDOI
01 Aug 1985
TL;DR: In this paper, a coaxially excited hollow probe in a rectangular waveguide has been analyzed by an approach in which the electric field off the end of the probe is the unknown that may be obtained by the solution of an integral equation.
Abstract: The problem of a coaxially excited hollow probe in a rectangular waveguide has been analysed by an approach in which the electric field off the end of the probe is the unknown that may be obtained by the solution of an integral equation. The electric field is represented as the static gap field multiplied by a Chebyshev series. The unknown coefficients may be found from a matrix equation in which the square matrix has dominant diagonal terms. It is shown that this permits accurate results to be obtained by retaining only a few terms of the series. Expressions are derived for the exterior and interior current distributions, and for the admittance presented to the coaxial line from which the probe is driven. Comparison of theoretical and experimental results shows the theory to be very accurate.

43 citations


Journal ArticleDOI
Jun Tomiyama1
TL;DR: In this article, the identity map, the transpose map, and the map τ(x)− 1 n (x) Tr (x)-1 n in the matrix algebra Mn are illustrated, providing parametric examples of maps which are not k+1-positive but which satisfy the matrix Schwartz inequality of order k for 1 ⩽ k⩽ n − 1.

40 citations


Journal ArticleDOI
Franz Rendl1
TL;DR: It is shown that decomposing a square matrix into a weighted sum of permutation matrices, such that the sum of the weights becomes minimal, is NP-hard.

37 citations


Journal ArticleDOI
TL;DR: This work proves constructively duality theorems of linear and quadratic programming in the combinatorial setting of oriented matroids and suggests the study of properties of square matrices such as symmetry and positive semi-definiteness in the context of orientedMatroids.

30 citations


Journal ArticleDOI
TL;DR: In this article, the authors present a general and rigorous solution for deconvolution with a deterministic time-varying and multichannel operator design, and demonstrate a straightforward least-squares error solution without simplifying to a Toeplitz matrix.
Abstract: A number of excellent papers have been published since the introduction of deconvolution by Robinson in the middle 1950s. The application of the Wiener‐Levinson algorithm makes deconvolution a practical and vital part of today’s digital seismic data processing. We review the original formulation of deconvolution, develop the solution from another perspective, and demonstrate a general and rigorous solution that could be implemented. By “general” we mean a deterministic time‐varying and multichannel operator design, and by “rigorous” we mean the straightforward least‐squares error solution without simplifying to a Toeplitz matrix. Also we show that the conjugate‐gradient algorithm used in conjunction with the least‐squares problem leads to a satisfactory simplification; that in the computation of the operators, the square matrix involved in the normal equations need not be computed. Furthermore, the product of this matrix with a column matrix can be obtained directly from the data as a result of two cascad...

Journal ArticleDOI
TL;DR: In this article, the congruence class of a Hermitian matrix is determined by the Oono-Imamura index of the pseudo-positive matrix function, and the connection with the matrix Cauchy index is established.
Abstract: The minimal realizations of pseudo-positive, pseudo-bounded, and pseudo-Schur rational matrix functions are constrained to satisfy certain matrix inequalities involving a Hermitian matrix. We show that the congruence class of this Hermitian matrix is determined by the Oono-Imamura index of the pseudo-positive matrix function. In the pseudo-lossless case, the connection with the matrix Cauchy index is established. A remarkable relationship between the realizations of a pseudo-positive matrix and the corresponding pseudo-Schur matrix is pointed out.

Journal ArticleDOI
TL;DR: The decomposition of fuzzy matrices is closely related to fuzzy databases and fuzzy retrieval models and some properties of decomposition are shown.
Abstract: A problem of decomposition of fuzzy rectangular matrices is examined and some properties of decomposition are shown. Any fuzzy matrix can be factored into a product of a square matrix and a rectangular matrix of the same dimension. This square matrix has reflexivity and transitivity. The decomposition of fuzzy matrices is closely related to fuzzy databases and fuzzy retrieval models.

Journal ArticleDOI
TL;DR: The asymptotic covariance matrix of the sample correlation matrix is derived in matrix form as an application of some new matrix theory in multivariate statistics as mentioned in this paper, where the covariance is derived as a function of the correlation matrix.

Journal ArticleDOI
TL;DR: In this article, the problems of solvability, controllability, and observability for the singular system Kx(t)=Ax(t)+Bu(t), where K is a singular, square matrix andu(t) is a complex vector function sufficiently differentiable, are studied.
Abstract: The problems of solvability, controllability, and observability for the singular systemKx(t)=Ax(t)+Bu(t) are studied, whereK is a singular, square matrix andu(t) is a complex vector function sufficiently differentiable. The classical theories of matrix pencils are first related to the solvability of singular systems. Then, the concepts of reachability, controllability, and observability of regular systems are extended to singular systems. Finally, the set of reachable states is described. The proposed matrix conditions for testing the controllability and observability of singular systems are simple and always feasible.

Journal ArticleDOI
TL;DR: A method in which the weight matrix can be decomposed into matrices of smaller order is proposed, which makes inverting the matrix computationally less heavy and extends the usefulness of ADF methods to applications with a larger number of variables.
Abstract: An important problem with asymptotic distribution-free (ADF) methods is the size of the weight matrix. Whereas under the assumption of normality of the observed variables the weight matrix can nicely be decomposed into two matrices of smaller order, under non-normality this cannot be done straightforwardly. In this paper we propose a method in which the weight matrix can be decomposed into matrices of smaller order, which makes inverting the matrix computationally less heavy and extends the usefulness of ADF methods to applications with a larger number of variables. An additional advantage of our method is that the weight matrix is formulated in terms of model parameters. As a consequence, one should expect the weight matrix to be more stable than in cases in which the weight matrix is computed from the data itself. In addition, estimates of the parameters may be less biased, a problem which often arises in ADF methods.

Posted Content
TL;DR: In this paper, the conditions under which unique differentiable functions λ(X) and u(X), respectively, exist in a neighborhood of a square matrix (complex or otherwise) satisfying the equations Xu = λu, λO, and Xu = ǫ.
Abstract: Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, . We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and . We obtain the first and second derivatives of λ(X) and the first derivative of u(X). Two alternative expressions for the first derivative of λ(X) are also presented. (This abstract was borrowed from another version of this item.)

Journal ArticleDOI
TL;DR: In this article, Cramer's inversion formula for the distribution of a quotient is generalized to matrix variates and applied to give an alternative derivation of the matrix t-distribution.

Journal ArticleDOI
TL;DR: In this paper, the authors give necessary and sufficient conditions for the validity of the inequality for every 0 ≤ λ ≤ 1, where A is a square matrix of random variables which is almost surely positive semi-definite.
Abstract: It is well-known that if A and B are two positive definite matrices of the same order and 0 ≤ λ ≤ 1, then . It is easy to construct an example consisting of two positive semi-definite matrices for which the above inequality is not true when one replaces the inverse operation by Moore-Penrose inverse operation. In this paper we give necessary and sufficient conditions for the validity of the inequality for every 0 ≤ λ ≤ 1. As an application, we give a sufficient condition under which the inequality (EA)+ λ E (A +) is valid, where A is a square matrix of random variables which is almost surely positive semi-definite, generalizing the well-known result (EA)− ≤ EA−1 when A is almost surely positive definite.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the conjecture is true and that if M is a square matrix with entries in a field of characteristic different from two, then M is expressible as a sum of two squares.
Abstract: Griffin and Krusemeyer have cinjecturted that if M is a square matrix with entries in a field of characteristic different from two and if M is not a scalar multiple of the indentity then M is expressible as a sum of two squares. In this paper it is shown that that conjecture is true.

Journal ArticleDOI
TL;DR: In this paper, necessary and sufficient conditions for such matrices to have a nonnegative Drazin inverse are presented, where the conditions are based on the nonnegative power of the matrix.

Journal ArticleDOI
TL;DR: In this article, a set of simultaneously triangularizable square matrices over an arbitrary field is considered, and it is shown that if the matrices are also quasicommutative, then they have a common eigenvector for every distinct set of corresponding eigenvalues.

Journal ArticleDOI
TL;DR: In this article, the structural zero of the inverse matrix of the Lyapunov equation has been studied in the context of structural transfer functions and the reachability matrix of any function of a matrix.
Abstract: Results are presented on the structural zeros of the inverse matrix. Applications are given relating to the Lyapunov equation, eigenvectors, structural transfer functions, the reachability matrix of any function of a matrix, and input-output reachability.

Journal ArticleDOI
TL;DR: In this paper, the photochemical acceleration of the thermal relaxation rate in the stratosphere was expressed as a matrix multiplication between a square matrix, which may be calculated for quite general conditions, and a vector representing the deviation of temperature from an equilibrium profile.
Abstract: It is shown that photochemical acceleration of the thermal relaxation rate in the stratosphere may be expressed as a matrix multiplication between a square matrix, which may be calculated for quite general conditions, and a vector representing the deviation of temperature from an equilibrium profile. Such a matrix is presented and its form discussed. Use of this matrix allows for temperature perturbations of any vertical scale and thus provides an accurate, as well as fast, method for calculating photochemical acceleration suitable for use in numerical models of stratospheric dynamics. the inclusion of the ozone 9.6 mm band into the heating rate calculations is shown to reduce the photochemical relaxation rate in the upper stratosphere.

Posted Content
01 Jan 1985
TL;DR: In this paper, the conditions under which unique differentiable functions λ(X) and u(X), respectively, exist in a neighborhood of a square matrix (complex or otherwise) satisfying the equations Xu = λu and Xu = ǫ.
Abstract: Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and We obtain the first and second derivatives of λ(X) and the first derivative of u(X) Two alternative expressions for the first derivative of λ(X) are also presented (This abstract was borrowed from another version of this item)

Journal ArticleDOI
TL;DR: In this article, the authors studied matrices of the form X =[ e e ij ], where x is an indeterminate defined over the rational field Q and there is a fascinating interplay between the combinatorial structures of the matrices E and X.

Book ChapterDOI
01 Jan 1985
TL;DR: In this article, the Moore-Penrose generalized inverse of a covariance matrix is expressed in Macsyma and examples of its use in statistical analysis are given, as well as the computer derivation of a novel form of the generalized inverse.
Abstract: Problems in data analysis, electrical networks, and finite element methods often involve linear models having singular, square matrices or matrices which are not square. The concept of generalized inverse extends the ranges of application of the notion of matrix inverse to such matrices and provides powerful mathematical tools for handling them. The use of symbolic generalized inverses during the analyses of these problems is equally powerful but requires more manipulation than can be reasonably performed by a human analyst. In this paper, the symbolic calculation of the Moore-Penrose generalized inverse, based on Albert’s limit formulation, will be expressed in Macsyma and examples of its use will be given. In particular, the computer derivation of a novel form of the generalized inverse of a covariance matrix typically used in statistical analysis will be shown.

Journal ArticleDOI
TL;DR: In this paper, the inequalities su At su Am  su Ap su Aq and At + su A ǫ + su Ap + Aǫ+ǫǫ plus su A +ǫ Gǫ are studied and generalized.
Abstract: The inequalities su At su Am  su Ap su Aq and At + su Am  su Ap + su Aq are studied and generalized. Here su A denotes the sum of elements of the square matrix A.

Dissertation
01 Jul 1985
TL;DR: The solution to theNearness to unitary and nearness to Hermitian positive (semi-) definiteness problems in terms of the polar decomposition is expressed and a quadratically convergent Newton iteration for computing the unitary polar factor is presented and analysed, and the iteration is developed into a practical algorithm for Computing the Polar decomposition.
Abstract: We consider the theoretical and the computational aspects of some nearness problems in numerical linear algebra. Given a matrix $A$, a matrix norm and a matrix property P, we wish to find the distance from $A$ to the class of matrices having property P, and to compute a nearest matrix from this class. It is well-known that nearness to singularity is measured by the reciprocal of the matrix condition number. We survey and compare a wide variety of techniques for estimating the condition number and make recommendations concerning the use of the estimates in applications. We express the solution to the nearness to unitary and nearness to Hermitian positive (semi-) definiteness problems in terms of the polar decomposition. A quadratically convergent Newton iteration for computing the unitary polar factor is presented and analysed, and the iteration is developed into a practical algorithm for computing the polar decomposition. Applications of the algorithm to factor analysis, aerospace computations and optimisation are described; and the algorithm is used to derive a new method for computing the square root of a symmetric positive definite matrix. This leads us, in the remainder of the thesis, to consider the theory and computation of matrix square roots. We analyse the convergence properties and the numerical stability of several well-known Newton methods for computing the matrix square root. By means of a perturbation analysis and supportive numerical examples it is shown that two of these Newton iterations are numerically unstable. The polar decomposition algorithm, and a further Newton square root iteration are shown not to suffer from this numerical instability. For a nonsingular real matrix $A$ we derive conditions for the existence of a real square root, and for the existence of a real square root which is a polynomial in $A$; the number of square roots of the latter type is determined. We show how a Schur method recently proposed by Bj\"orck and Hammarling can be extended so as to compute a real square root of a real matrix in real arithmetic. Finally, we investigate the conditioning of matrix square roots and derive an algorithm for the computation of a well-conditioned square root.

Journal ArticleDOI
TL;DR: The spectral radius of a complex square matrix A is given by ρ ( A ) = lim sup k → ∞ (Tr A k ) 1/k as discussed by the authors, which gives information about the moduli of all eigenvalues of A.