scispace - formally typeset
Search or ask a question

Showing papers on "Square matrix published in 1982"


Proceedings ArticleDOI
30 Jul 1982
TL;DR: In this paper, a unified concept of using systolic arrays to perform real-time triangularization for both general and band matrices is presented, and a framework is presented for the solution of linear systems with pivoting and for least squares computations.
Abstract: Given an n x p matrix X with p < n, matrix triangularization, or triangularization in short, is to determine an n x n nonsingular matrix Al such that MX = [ R 0 where R is p x p upper triangular, and furthermore to compute the entries in R. By triangularization, many matrix problems are reduced to the simpler problem of solving triangular- linear systems (see for example, Stewart). When X is a square matrix, triangularization is the major step in almost all direct methods for solving general linear systems. When M is restricted to be an orthogonal matrix Q, triangularization is also the key step in computing least squares solutions by the QR decomposition, and in computing eigenvalues by the QR algorithm. Triangularization is computationally expensive, however. Algorithms for performing it typically require n3 operations on general n x n matrices. As a result, triangularization has become a bottleneck in some real-time applications.11 This paper sketches unified concepts of using systolic arrays to perform real-time triangularization for both general and band matrices. (Examples and general discussions of systolic architectures can be found in other papers.6.7) Under the same framework systolic triangularization arrays arc derived for the solution of linear systems with pivoting and for least squares computations. More detailed descriptions of the suggested systolic arrays will appear in the final version of the paper.© (1982) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

474 citations


Journal ArticleDOI
TL;DR: It is shown that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L, S, and P can be found in O(mα−1 n) time, where the complexity of matrix multiplication is O( mα).

168 citations


Journal ArticleDOI
TL;DR: The controversy that exists in the literature concerning the solutions of such equations is investigated in this paper, where the authors arrive at solutions through an application of singular perturbation theory, and the controversy is discussed.

90 citations


Journal ArticleDOI
TL;DR: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0.
Abstract: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilbert's projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated m...

83 citations


Book ChapterDOI
01 May 1982
TL;DR: In this paper, the authors considered classical links of two components and showed how to obtain a pair of linking forms from the analogue of a Seifert surface, and from these the first homology of the universal abelian (ℤ ⊕ ℤ) cover is obtained, thus giving a practical method for calculating the Alexander polynomial.
Abstract: Introduction Given a Seifert surface for a classical knot, there is associated a linking form from which the first homology of the infinite cyclic cover may be obtained. This article considers classical links of two components and shows how to obtain a pair of linking forms from the analogue of a Seifert surface. From these the first homology of the universal abelian (ℤ ⊕ ℤ) cover is obtained, thus giving a practical method for calculating the Alexander polynomial. Also obtained is a new signature invariant for links. The method generalises to links of any number of components; however this is not done here. In this paper, unless otherwise stated, a link will mean a piecewise-linear embedding of two oriented circles in the three sphere S 3 . The main results are (2.1) and (2.4). The former provides a square matrix presenting the first homology of the cover obtained from the Hurewicz homomorphism of the link complement. The latter gives a signature invariant, obtained from this matrix, which vanishes for strongly slice links. On the way some known results are obtained, namely Torres' conditions on a link polynomial, and a result of Kawauchi and independently Nakagawa on the (reduced) Alexander polynomial of a strongly s1ice 1ink. The paper is organised as follows. Section 2 contains the method of obtaining the matrix used in (2.1) and states the main results. The reader not interested in the proofs need read no further.

49 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present an algorithm which when applied to a real square matrix A gives a definite yes or no answer to the question: given A, does there exist a positive diagonal matrix P such that PA + A T + P is negative definite?
Abstract: Given a real square matrix A we address the question: Does there exist a positive diagonal matrix P such that PA + A^{T}P is negative definite? We present an algorithm which when applied to A gives a definite yes or no answer to this question. If the answer is yes, the algorithm will provide one such P .

48 citations


Journal ArticleDOI
TL;DR: In this article, the authors extend the use of Dwyer's formulas to symmetric matrices in which the (i, j) element is considered functionally equal to the (j, i) element.
Abstract: Dwyer (1967) provided extensive formulas for matrix derivatives, many of which are for derivatives with respect to symmetric matrices. The results of his article are only for symmetric matrices whose (j, i) element is considered to differ from the (i, j) element even though their scalar values are equal. A simple result is given that extends the use of Dwyer's formulas to symmetric matrices in which the (i, j) element is considered functionally equal to the (j, i) element. Application of the results are illustrated by deriving the multivariate normal information matrix.

40 citations


Journal ArticleDOI
TL;DR: In this article, the shorting of an operator was extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite, and some applications of the shorted matrix in mathematical statistics were discussed.

34 citations


Proceedings Article
01 Jan 1982
TL;DR: A concurrent algorithm and corresponding array for computing the triangular matrix R by Householder transformations is described and particular attention is given to issues such as broadcasting and pipelining.
Abstract: The QR-method is a method for the solution of linear system of equations. The matrix R is upper triangular and Q is a unitary matrix. In equation solving Q is not always computed explicitly. The matrix R can be obtained by applying a sequence of unitary transformations to the matrix defining the system of equations. Householder's method or Given's method can be used to determine unitary transformation matrices. This paper describes a concurrent algorithm and corresponding array for computing the triangular matrix R by Householder transformations. Particular attention is given to issues such as broadcasting and pipelining.

31 citations


Journal ArticleDOI
TL;DR: The main purpose of this correspondence is to establish an expression for the solution of the Lyapunov matrix equation in terms of the principal indempotents and nilpotents of the coefficient matrices.
Abstract: The main purpose of this correspondence is to establish an expression for the solution of the Lyapunov matrix equation in terms of the principal indempotents and nilpotents of the coefficient matrices. The coefficient matrices are not necessarily of the same size and may have common characteristic roots and have elements belonging to the field of complex numbers.

28 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that every real matrix contained in a matrix interval is sign-regular if and only if two special matrices taken from that matrix interval are sign regular.

Proceedings Article
01 Jan 1982
TL;DR: Quadratic filters are the simplest non linear time-invariant systems and correspond to the second term of the Volterra expansion and their kernel is a symmetric finite or infinite square matrix.
Abstract: Quadratic filters are the simplest non linear time-invariant systems and correspond to the second term of the Volterra expansion. Such filters are completely defined by their kernel which is a symmetric finite or infinite square matrix. Some examples are presented. The harmonic representation of such filters is discussed and also the calculation of the statistical properties of the output in the case of a Gaussian input. Finally some problems of implementation of such filters are discussed.

Journal ArticleDOI
TL;DR: In this article, a transformation matrix that transforms a right (left) solvent to the corresponding left (right) solvent of a matrix polynomial has been derived, which is a block partial fraction expansion of a rational matrix.
Abstract: The main contribution. is a block partial fraction expansion of a rational matrix are polynomial matrices. A new algorithm is derived to construct a transformation matrix that transforms a right (left) solvent to the corresponding left (right) solvent of a matrix polynomial. Also, the algorithm can be used to construct a set of right (left) fundamental matrix polynomials and the inversion of a block Vandermonde matrix. This leads to a new technique to perform the block partial fraction expansion of a class of rational matrices.

Journal ArticleDOI
TL;DR: In this paper, the compuation of the characteristic polynomial of a square matrix is reviewed and a new method of its computation is suggested, which has potential application in the design of state estimators and state feedbacks.
Abstract: In this note, the compuation of the characteristic polynomial of a square matrix is reviewed. A new method of its computation is then suggested. The suggested method has potential application in the design of state estimators and state feedbacks.

Journal ArticleDOI
S. Ahn1
TL;DR: In this paper, the determinant of a nonsingular real (m \times m ) matrix polynomial of n th order has all its roots inside the unit circle.
Abstract: Two sufficient conditions that the determinant of a nonsingular real ( m \times m ) matrix polynomial of n th order has all its roots inside the unit circle have been obtained. These conditions are represented in terms of rational functions of the coefficient matrices. Therefore, these conditions do not require the computation of the determinant polynomial. The first condition is given in terms of the positive definiteness of an ( mn \times mn ) symmetric matrix, while the second condition is expressed by the positive definiteness of an ( m \times m ) Hermitian matrix which is a function of z, |z| \leq 1 . The first condition implies the second, and hence is more restrictive than the second.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a non-invertible totally positive matrix A has one and only one "main diagonal", which is the property that all finite sections of A principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to A I.

01 Jan 1982
TL;DR: In this article, a method is developed to determine optimal iteration parameters for use in a complex version of Richardson's iteration for a spectrum contained in any simply-connected bounded open set (in practice, a polygon symmetric with respect to the real axis).
Abstract: In 1975, T. A. Manteuffel developed a method for the iterative solution of a non-symmetric linear system, Ax = b, when the eigenvalues have positive real parts. The iterative parameters are reciprocals of the roots of a scaled and translated Chebyshev polynomial and depend upon an ellipse enclosing the spectrum of the system matrix. In some applications, a matrix will occur whose spectrum is not well-approximated by an ellipse. In this thesis, a method is developed to determine optimal iteration parameters for use in a complex version of Richardson's iteration for a spectrum contained in any simply-connected bounded open set (in practice, a polygon symmetric with respect to the real axis). The proposed method is a generalization of algorithms found in a 1958 paper of E. L. Stiefel in which real orthogonal polynomials were used. In this thesis, Stiefel's work is extended to complex orthogonal and bi-orthogonal polynomials. In addition, numerical methods are developed to obtain the desired iteration parameters by means of least squares theory. Results are presented which show that the methods of this thesis compare favorably to Manteuffel's method and the Lanczos algorithm for test matrices whose spectra had pre-determined shapes.

Journal ArticleDOI
TL;DR: In this paper, a simple algorithm is presented for testing the diagonal similarity of two square matrices with entries in a field, which is based on the existence of a canonical form for diagonal similarity.
Abstract: A simple algorithm is presented for testing the diagonal similarity of two square matrices with entries in a field. Extended forms of the algorithm decide various related problems such as the simultaneous diagonal similarity of two families of matrices, the existence of a matrix in a subfield diagonally similar to a given matrix, the existence of a unitary matrix similar to a given complex matrix, and the corresponding problems for diagonal equivalence in place of diagonal similarity. The computational complexity of our principal algorithm is studied, programs and examples are given. The algorithms are based on the existence of a canonical form for diagonal similarity. In the first part of the paper theorems are proved which establish the existence of this form and which investigate its properties.

Journal ArticleDOI
TL;DR: The solution for the finite-time matrix Riccati equation is presented and it is shown that the solution is obtained in terms of the partition of the transition matrix.
Abstract: The solution for the finite-time matrix Riccati equation is presented in this paper. The solution to the Riccati equation is obtained in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and solved using Laplace transforms and the computation is done by the digital computer. A numerical example for the proposed method is given.

01 Jan 1982
TL;DR: In this paper, the authors generalized linear matrix equations to nonsquare equations, nonpolynomial type equations, in particular equations given by an integr~l, and finally to equations over an arbitrary commutative ring with unit element.
Abstract: Linear matrix equations were studied by Sylvester, Stephanos, Datuashvili and Roth. In this paper, solvability tz::Onditions given by. these authors are generalized in various directions: to nonsquare equations, nonpolynomial type equations, in particular equations given by an integr~l, and finally to equations over an arbitrary commutative ring with unit element.


Journal ArticleDOI
TL;DR: In this article, the role of λ = 1 as a root of the minimal polynomial of a positive n × n matrix with spectral radius r ( A ) = 1 was discussed.

Journal ArticleDOI
TL;DR: For every real square matrix A there exists a nonsingular real matrix X, for which X − 1 AX is a tridiagonal matrix, and under certain conditions X is uniquely determined.
Abstract: For every real square matrix A there exists a nonsingular real matrix X, for which X −1 AX is a tridiagonal matrix, and under certain conditions X is uniquely determined. Simple proofs are presented for the existence and uniqueness of this transformation.

Journal ArticleDOI
TL;DR: A time-optimal algorithm for finding the cyclic index of an irreducible nonnegative square matrix, which is the number of eigenvalues of maximum modulus of that matrix if the matrix is said to be primitive.
Abstract: The cyclic index $\delta $ of an irreducible nonnegative square matrix is the number of eigenvalues of maximum modulus of that matrix. If $\delta = 1$, the matrix is said to be primitive. The notions of primitivity and cyclic index play an important role in the theory of nonnegative matrices. In the context of discrete Markov chains, the words “period” and “aperiodic” are sometimes used in place of “cyclic index” and “primitive”, respectively. It is known how to test an irreducible nonnegative square matrix for primitivity, but there is no known practical method for finding the cyclic index $\delta $ in the general case. This paper presents a time-optimal algorithm for finding $\delta $.

Journal ArticleDOI
F. Incertis1
TL;DR: In this paper, the authors considered the 2-D inverse filtering problem with the discrete Fourier transform (DFT) method and derived the convergence conditions of the DFT under periodicity assumptions.
Abstract: The linear and space-invariant convolution Y = H \bigotimes X of a matrix X with the kernel (impulse response, point spread function) matrix H is modeled as a linear matrix equation of the form Y=\sum_{i}A_{i}XB_{i} . Under periodicity assumptions on matrix X analytical and numerical solution methods of the 2-D inverse filtering problem are presented jointly with an analysis of the computational complexity and convergence conditions. As a consequence of this new matrix approach, the discrete Fourier transform (DFT) method is derived as a particular case of more powerful algebraic operators to solve general matrix equations.


Journal ArticleDOI
TL;DR: This paper is primarily concerned with the derivation of four theorems involving the set intersection matrix Y, which reveal the extent to which the polynomial det(Y) and the characteristic polynometric f(z) of Y characterize the set intersections matrix Y.

Journal ArticleDOI
TL;DR: In this article, the Lyapunov matrix equation is used to obtain the estimate for the transient behavior of linear constant systems, and it is shown that the equation gives information not only on the maximum real part of the characteristic roots of the system matrix but also on other extremal values to these roots.
Abstract: It is well known that the Lyapunov matrix equation can be utilized to obtain the estimate for the transient behavior of linear constant systems. This paper shows that the equation gives information not only on the maximum real part of the characteristic roots of system matrix but also on other extremal values to these roots.


Journal ArticleDOI
TL;DR: In this article, the residue theorems of rational A-matrices having complex variables (A) and the associated rational matrix functions with square matrix variables were derived by employing the generalized Cauchy' s integral theorem.
Abstract: This paper deals with the residue theorems of rational A-matrices having complex variables ( A) and the associated rational matrix functions with square matrix variables. First, some fundamental properties of A-matrices and the associated matrix poly. normals with square matrix variables are defined by employing the Cauchy' s integral theorem. Then, a new matrix residue theorem is derived via the generalized Cauchy' s integral theorem. Finally, the block partial fraction expansion and spectral factorization of rational A-matrieos are developed via the newly extended matrix residue theorem.