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Showing papers on "Square matrix published in 1983"


Journal ArticleDOI
TL;DR: This paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.

364 citations


Journal ArticleDOI
TL;DR: Various representations for systems described by aset of high-order differential equations of the form R0w + R1w + … + Rsw(s) = 0, with R0, R1,…, Rs not necessarily square matrices are developed.

119 citations


Journal ArticleDOI
TL;DR: In this article, conditions for the existence of solutions of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal).
Abstract: Conditions for the existence of solutions, and the general solution of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal). This theory is subsequently extended to matrix equations that are linear in several unknown ‘structured’ matrices, and to partitioned matrix equations.

72 citations


Posted Content
TL;DR: In this paper, conditions for the existence of solutions of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal).
Abstract: Conditions for the existence of solutions, and the general solution of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal). This theory is subsequently extended to matrix equations that are linear in several unknown ‘structured’ matrices, and to partitioned matrix equations.

69 citations


Journal ArticleDOI
TL;DR: In this article, a method for calculating the eigenvalues of the transfer matrix is proposed based on some fundamental properties of the R matrix such as unitarity, cross symmetry, and special relations related to the structure of the degeneracy of R matrix at certain points.
Abstract: The method proposed for calculating the eigenvalues of the transfer matrix is based on some fundamental properties of the R matrix such as unitarity, cross symmetry, and special relations related to the structure of the degeneracy of the R matrix at certain points. One can therefore hope that it applies to a large class of models in which the complicated structure of the R matrix makes it impossible to construct n-particle eigenstates of the transfer matrix by the Bethe ansatz.

46 citations


Journal ArticleDOI
TL;DR: An analog of the characteristic polynomial is defined for a matrix over the algebraic structure ( R, max, +) and its properties are discussed.

42 citations


Journal ArticleDOI
TL;DR: A simple and constructive proof for the existence of a real symmetric matrix with prescribed diagonal elements and eigenvalues is given in this article, where numerical implementable algorithms for constructing such a matrix are discussed.

42 citations


Journal ArticleDOI
TL;DR: How computing time can be gained in the important symmetric diagonal dominant case if only part of the diagonal is needed is indicated and computing times of the method of Takahasi et al.6 with several variants of columnwise inversion are compared.
Abstract: The Gauss-Jordan inversion algorithm requires 0(n3) arithmetic operations. In some practical applications, like state estimation or short-circuit calculation in power systems, the given matrix is sparse and only part of the inverse is needed. Most frequently in the diagonal dominant case this is the diagonal. There are two ways to exploit sparsity to determine elements of the inverse: 1Columnwise inversion via the solution of sparse linear systems with columns of the unit matrix as right-hand sides. 2Application of the algorithm of Takahashi et al6 . The latter algorithm arises very naturally from multiplying the left- and right-hand factors of the Zollenkopf bifactorization in reverse order. We indicate how computing time can be gained in the important symmetric diagonal dominant case if only part of the diagonal is needed and compare computing times of the method of Takahasi et al.6 with several variants of columnwise inversion. Whereas most of the theory holds for general matrices, experiments are performed on the symmetric diagonal dominant case. For band matrices the operation count is 0(n) both for computing a column of the inverse by columnwise inversion and the diagonal by the algorithm of Takahashi et al6.

33 citations


Journal ArticleDOI
TL;DR: It is shown that this type of matrices can be decomposed into the Hadamard Product of a consistent matrix and an inconsistent matrix, which is used in the analysis of sensitivity to compute the principle eigenvector of a perturbed reciprocal matrix.

32 citations


Journal ArticleDOI
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


Journal ArticleDOI
01 Dec 1983
TL;DR: In this paper, it was shown that the coefficients of linear prediction for a random process and the prediction error variances are related to the covariance matrix through triangular decomposition, and that the matrix is written in the product form LDL* where L is lower triangular with unit diagonal and D is diagonal.
Abstract: It is shown that the coefficients of linear prediction for a random process and the prediction error variances are related to the covariance matrix through triangular decomposition. In particular, if the covariance matrix is written in the product form LDL*where L is lower triangular with unit diagonal and D is diagonal, then the rows of L-1are the coefficients of linear prediction and the elements of D are the prediction error variances.

Journal ArticleDOI
TL;DR: In this paper, a modified propagator matrix method was proposed for the calculation of the two-dimensional elasto-dynamic Green's function for a stratified medium, where the solution is represented in the form of an inverse Fourier integral which is to be integrated along a properly chosen path in the complex wavenumber plane.
Abstract: Summary. The calculation of the two-dimensional elasto-dynamic Green’s function for a stratified medium is investigated. The solution is represented in the form of an inverse Fourier integral which is to be integrated along a properly chosen path in the complex wavenumber plane. The integrand is computed using a modified propagator matrix method. This method is based on a mixed formulation using the propagator matrix and the matrix of minors of the propagator matrix (compound matrix). The major advantages of this approach are the elimination of the numerical loss of precision problems associated with the Thomson-Haskell formulation, without losing the attractive tractability and compactness of the propagator matrix method. This modified method is first mathematically derived, and theoretical seismograms are then presented for two examples.

Journal ArticleDOI
TL;DR: In this article, the authors propose to decompose a symmetric optical system into the product of two upper triangular matrices and an antidiagonal matrix for propagation between any two arbitrary planes, and the decomposition yields the above-mentioned focal-plane matrix.
Abstract: In the paraxial approximation a symmetrical optical system may be represented by a 2 × 2 matrix. It has been the custom to describe each optical element by a transfer matrix representing propagation between the principal planes or through an interface for thin elements. If the focal-plane representation is used instead, any focusing element or combination of elements is represented by the same antidiagonal matrix whose nonzero elements are the focal lengths: The matrix represents propagation between the focal planes. For propagation between any two arbitrary planes, the system transfer matrix can be decomposed into the product of two upper triangular matrices and an antidiagonal matrix. This decomposition yields the above-mentioned focal-plane matrix, and the two upper triangular matrices represent propagation between the input and the output planes and the focal planes. Because the matrix decomposition directly yields the parameters of interest, the analysis and the synthesis of optical systems are simpler to carry out. Examples are given for lenses, diopters, mirrors, periodic sequences, resonators, lenslike media, and phase-conjugate mirror systems.

Journal ArticleDOI
TL;DR: In this paper, a unified spectral method for the analysis of regular matrix polynomials with non-zero determinant was proposed, which extends the above methods and results to the class of all regular matrix regular polynomial matrices.
Abstract: INTRODUCTION A spectral method for the analysis of monic matrix polynomials was developed recently in [8-10]. A similar method was developed [11-12] for non-monic, especially comonic, matrix polynomials. The present work suggests a unified approach, which extends the above methods and results to the class of all regular matrix polynomials (i.e. with non-zero determinant). The spectral method basically concerns the construction of a pair of matrices (X,T) , called a standard pair, from the spectral data of the given matrix polynomial A(I) , and is described in Sections 1-4. The inverse problem, namely the construction of A(1) from the pair (X,T] , is described in Section 5. In Section 6 we discuss a representation for A-l(1) and its application to basic linear systems of equations, and in Section 7, all factorizations of A(I) are characterized by certain restrictions on the pair (X,T) .

Journal ArticleDOI
TL;DR: De Sa and Thompson as mentioned in this paper characterized the relationship between the similarity invariants of a square matrix over a field and those of a principal submatrix of that matrix, and the results on eigenvalues of complex hermitian matrices (the Courant-Fischer minimax theorem and the Cauchy interlacing theorem), on singular values of rectangular complex matrices and on invariant factors of rectangular matrices over a principal ideal domain.

Journal ArticleDOI
TL;DR: The theory of rational matrix functions W(λ) = = C(λI-A)−1B + D, as presented in the book Bart-Gohberg-Kaashoek [1], is extended here to the case where D = W(∞) is not invertible, and applied to factorizations of monic matrix polynomials.
Abstract: The theory of factorization of rational matrix functions W(λ) = = C(λI-A)−1B + D, as presented in the book Bart-Gohberg-Kaashoek [1], is extended here to the case where D = W(∞) is not invertible, and applied to factorizations of monic matrix polynomials.

Journal ArticleDOI
TL;DR: In this paper, the authors give a method for determining the exponential generating function for the coefficient of p-regular labeled graphs and square matrices with row and column sums equal to p. This generating function is called the Hammond series of S and they use it to show that the counting series for certain combinatorial problems satisfy linear recurrence equations with polynomial coefficients.
Abstract: We give a method for determining the exponential generating function for the coefficient of $x_1^P \cdots x_n^P $ in a symmetric function S in the indeterminates $x_1 , \cdots ,x_n $. This generating function is called the Hammond series of S, and we use it to show that the counting series for certain combinatorial problems satisfy linear recurrence equations with polynomial coefficients. These problems include p-regular labelled graphs and square matrices with row and column sums equal to p.

Journal ArticleDOI
TL;DR: In this paper, a description of inertia characteristics applicable to both a self-adjoint matrix polynomial and an associated perturbed matrix is given in terms of the inertia of a Bezout matrix and a Hankel matrix, and applications of the results include lower bounds for the number of real eigenvalues, skew perturbations, and a problem concerning controllability and linear vibrations.

Journal ArticleDOI
TL;DR: In this paper, the spectral properties of the matrix seminorm f(B) were investigated for the case of a stochastic matrix B, which can be easily generalized to the nonnegative matrix B.
Abstract: One investigates estimates of the type ∥ABx∥⩽f(B)∥Ax∥, where A, B are matrices and x is a vector belonging to a certain subspace. One investigates the properties of the matrix seminorm f(B), in particular, its relation to the spectrum of the matrix B. For the case of a stochastic matrix B (which can be easily generalized to the case of a nonnegative matrix B) one derives estimates for f(B) which are convenient for practical computations (also on an electronic computer). One gives a numerical example illustrating the application of the results.

Journal ArticleDOI
TL;DR: In this article, the authors describe the method of expansion by diagonal elements for the evaluation of the determinant of the sum of two square matrices, one of which is diagonal, applied to the problem of obtaining the characteristic polynomial of a square matrix.
Abstract: This article describes the method of expansion by diagonal elements for the evaluation of the determinant of the sum of two square matrices, one of which is diagonal. This result is applied to the problem of obtaining the characteristic polynomial of a square matrix. For problems of interest in statistics, this technique is frequently much more direct than the usual approach of simplification (e.g., triangularization) by elementary row and column operations. Several examples are presented.

Journal ArticleDOI
TL;DR: In this paper, the transfer impedance in the Duffin-morley general linear electromechanical system has been studied and the results are used to study transfer impedance of a generalized inverse of the complex matrix V of order n×n.
Abstract: AMS (MOS)'s: Primary: 15A09 Secondary: 15A99 A complex matrix V of order n×n is said to be almost definite if for a complex n-tuple . Interrelations between the blocks of a generalized inverse of the complex matrix are obtained when V is an almost definite matrix. These results are used to study the transfer impedance in the Duffin-Morley general linear electromechanical system.

Journal ArticleDOI
TL;DR: In this paper, a new parametrization of the rotation matrix is presented assuming it is known that this matrix transforms a particular vector into another vector, and the matrix is expressed as a function of these vectors and of an angle which measures the deviation from the shortest possible rotation.
Abstract: A new parametrization of the rotation matrix is presented assuming it is known that this matrix transforms a particular vector into another vector. The matrix is expressed as a function of these vectors and of an angle which measures the deviation from the shortest possible rotation. Some applications of this representation to dynamics are also explained.

Journal ArticleDOI
TL;DR: Estimations of covariance matrices of the random disturbances, including the state noise, observation noise and random part in transfer matrix, associated with this class of stochastic bilinear systems are presented.
Abstract: This paper considers the identification of a class of stochastic bilinear systems in which the transfer matrix in state equation is the sum of a constant matrix A and a random matrix S(i), and observation equation is linear. Estimations of covariance matrices of the random disturbances, including the state noise, observation noise and random part in transfer matrix, associated with this system are also presented. Strong consistency of the estimations is also established.


Journal ArticleDOI
TL;DR: In this article, it was shown that the Moore-Penrose inverse A+ of a singular square matrix A does not for all regular matrices T satisfy the covariance condition (TAT−1)+ = TA+T−1.

Journal ArticleDOI
TL;DR: In this paper, a crystallographic interpretation of the elementary divisor theorem applied to square integral matrices is presented and the concept of a least multiplier of a rational square matrix is introduced and its properties are derived from the theorem.
Abstract: A crystallographic interpretation of the elementary divisor theorem applied to square integral matrices is presented. The concept of a least multiplier of a rational square matrix is introduced and its properties are derived from the theorem. These topics are relevant in the formulation of a general theory of coincidence-site lattices.

Journal ArticleDOI
TL;DR: In this article, a new method was proposed to construct the characteristic polynomial for a class of square matrices with some restrictions, and an algorithm was also given for computing it.
Abstract: In this correspondence, a new method is proposed to construct the characteristic polynomial for a class of square matrices with some restrictions. An algorithm is also given. The proposed method and algorithm have recursive forms and are therefore suitable for calculation by computer.


Journal ArticleDOI
Gábor Révész1
TL;DR: In this article, the authors considered the problem of ordering an R-field in terms of matrix cones over R rather than ordinary cones of elements of K. They gave necessary and sufficient conditions for such an orderable matrix cone over R to be orderable.
Abstract: We are concerned with the following problem. Given a ring R and an epic R-field K, under what conditions can K be fully ordered? Epic R-fields can be constructed in terms of matrices over R; this makes it natural to consider matrix cones over R rather than ordinary cones of elements of K. Essentially, a matrix cone over R, associated with a given ordering of K, consists of all square matrices which either become singular or have positive Dieudonne determinant over K. We give necessary and sufficient conditions in terms of matrix cones for (i) an epic R-field to be orderable, (ii) a full order on R to be extendable to a field of fractions of R and (iii) for such an extension to be unique.

Journal ArticleDOI
Ingram Olldn1
TL;DR: In this article, it was shown that the least squares estimate of a regression matrix converges with probability one to the population matrix, which is a matrix generalization of a quadratic inequality for real numbers.