Showing papers on "Square matrix published in 1987"
TL;DR: A special class of indefinite quadratic programs is constructed, with simple constraints and integer data, and it is shown that checking (a) or (b) on this class is NP-complete.
Abstract: In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether(a)a given feasible solution is not a local minimum, and(b)the objective function is not bounded below on the set of feasible solutions.
We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.
1,117 citations
TL;DR: An extension of the Schur method is presented which enables real arithmetic to be used throughout when computing a real square root of a real matrix.
208 citations
01 Feb 1987
TL;DR: In this paper, the authors discuss algorithms for matrix multiplication on a concurrent processor containing a two-dimensional mesh or richer topology, and present detailed performance measurements on hypercubes with 4, 16, and 64 nodes.
Abstract: We discuss algorithms for matrix multiplication on a concurrent processor containing a two-dimensional mesh or richer topology. We present detailed performance measurements on hypercubes with 4, 16, and 64 nodes, and analyze them in terms of communication overhead and load balancing. We show that the decomposition into square subblocks is optimal C code implementing the algorithms is available.
200 citations
TL;DR: In this paper, it is shown how to obtain a best approximation of lower rank in which a specified set of columns of the matrix remains fixed, and some applications of the generalization are discussed.
197 citations
TL;DR: Four new approaches are presented to the solution of those problems having several applications to combinatorial computations in order to extend the suboptimum time and processor bounds of (Pan and Reif, 1987) to the case of computing the inverse, determinant, and characteristic polynomial of an arbitrary integer input matrix.
99 citations
TL;DR: In this paper, the group inverse of a matrix is used to define the #-order on square matrices of index 1, which is similar to the ∗-order of Drazin [2] and the minus order of Hartwig [6, 10] and Nambooripad [17].
84 citations
TL;DR: In this paper, some existing stability and instability conditions of the second-order matrix polynomial were summarized and sufficient conditions for stability or instability were developed via the relevant linear matrix equation obtained from the Lyapunov theory.
Abstract: This note summarizes some existing stability and instability conditions of the second-order matrix polynomial which arises in the formulation of classical mechanics, aerodynamics, and robotic systems. Also, some sufficient conditions for stability or instability of the second-order matrix polynomial are newly developed via the relevant linear matrix equation obtained from the Lyapunov theory.
67 citations
TL;DR: A simple computer‐oriented method is presented for constructing the (molecular) distance matrix, where the distance matrix considered is the graph‐theoretical (topological)distance matrix.
Abstract: A simple computer-oriented method is presented for constructing the (molecular) distance matrix. The distance matrix considered is the graph-theoretical (topological) distance matrix.
61 citations
TL;DR: In this article, it was shown that if a matrix is represented in those coordinates where its closest normal matrix is diagonal, its restriction to any pair of coordinate directions is a multiple of a real diagonal and skew noniagonal 2×2 matrix.
Abstract: A method of finding the closest normal matrix in the Frobenius matrix norm is developed. It is shown that if a matrix is represented in those coordinates where its closest normal matrix is diagonal, its restriction to any pair of coordinate directions is a multiple of a real diagonal and skew nondiagonal 2×2 matrix. A convergent algorithm to bring an arbitrary matrix into that form is described and results of numerical tests are reported.
57 citations
TL;DR: In this article, a new proof for the inequality tr (XY) \leq \parallel X ∈ {2} \cdot tr Y under the condition that X may be any square matrix.
Abstract: A new proof is presented for the inequality, tr (XY) \leq \parallel X \parallel_{2} \cdot tr Y . This argument is valid under the condition that Y be real symmetric nonnegative definite; X may be any square matrix.
52 citations
TL;DR: An algorithm for producing a nonconditional simulation by multiplying the square root of the covariance matrix by a random vector by being evaluated as a convolution using the fast Fourier transform.
Abstract: An algorithm for producing a nonconditional simulation by multiplying the square root of the covariance matrix by a random vector is described. First, the square root of a matrix (or a function of a matrix in general) is defined. The square root of the matrix can be approximated by a minimax matrix polynomial. The block Toeplitz structure of the covariance matrix is used to minimize storage. Finally, multiplication of the block Toeplitz matrix by the random vector can be evaluated as a convolution using the fast Fourier transform. This results in an algorithm which is not only efficient in terms of storage and computation but also easy to implement.
TL;DR: In this paper, a finitely computable graph-theoretic answer is given to the following question concerning linear dynamical systems: When, given only the signs of entries (+, -, or 0) in a real square matrix A, can one be certain that all positive trajectories of the system ẋ = Ax are bounded?
DOI•
01 Apr 1987TL;DR: In this paper, the idea of computing a projection matrix for bearing estimations by using any M rows of the covariance matrix, where M is the number of sources, is extended to the cases of low signal/noise ratio (SNR) and totally correlated sources.
Abstract: In the paper the idea of computing a projection matrix for bearing estimations by using any M rows of the covariance matrix, where M is the number of sources, is extended to the cases of low signal/noise ratio (SNR) and totally correlated sources. For low SNR, the projection matrix can be obtained by using off-diagonal terms of M rows of the covariance matrix, and a reduced search vector can be used. For totally correlated sources, the idea of spatial averaging is adapted to compute the projection matrix. However, no averaging is actually carried out.
TL;DR: A necessary and sufficient condition for the existence of a square matrix with prescribed eigenvalues and prescribed complementary principal blocks is given in this paper. But this condition is not applicable to square matrices.
TL;DR: In this paper, the authors considered the case when G = SL(n+k,C) and G0=P(k) is a maximal parabolic subgroup of G, leaving a k-dimensional vector space invariant (1≤k≤n).
Abstract: A system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every Lie group–subgroup pair G⊇G0. We consider the case when G=SL(n+k,C) and G0=P(k) is a maximal parabolic subgroup of G, leaving a k‐dimensional vector space invariant (1≤k≤n). The nonlinear ordinary differential equations (ODE’s) in this case are rectangular matrix Riccati equations for a matrix W(t)∈Cn×k. The special case n=rk (n,r,k∈N) is considered and a superposition formula is obtained, expressing the general solution in terms of r+3 particular solutions for r≥2, k≥2. For r=1 (square matrix Riccati equations) five solutions are needed, for r=n (projective Riccati equations) the required number is n+2.
TL;DR: In this article, convergence proofs for one-sided Jacobi/Hestenes methods for the singular value problem are given, where the limiting form of the matrix iterates for the Hestenes method with optimization when the original matrix is normal.
Abstract: Convergence proofs are given for one-sided Jacobi/Hestenes methods for the singular value problem. The limiting form of the matrix iterates for the Hestenes method with optimization when the original matrix is normal is derived; this limiting matrix is block diagonal, where the blocks are multiples of unitary matrices. A variation in the algorithm to guarantee convergence to a diagonal matrix for the symmetric eigenvalue problem is shown. Implementation techniques for parallel computation, in particular, on the hypercube are indicated.
TL;DR: In this article, a fixed point theorem is used to show that a solution of Griffiths' equation (G) exists and the same proof shows that the eigenvalue @l is unique.
TL;DR: In this article, the strong law of large numbers for a multivariate martingale normed by a sequence of square matrices is considered for linear regression with multivariate responses.
TL;DR: In this paper, a new algorithm for computing matrix partial fractions representing the inverse of linear matrix pencils is presented, which is suitable to determine the matrix transfer function, and is computer oriented because all manipulations can be performed on matrices with constant entries.
Abstract: A new algorithm for computations of matrix partial fractions representing the inverse of linear matrix pencil is based on an appropriate expression in matrix form of the Pascal triangle. It concerns singular and nonsingular systems and starts with the inverse of regular matrix linear pencil M(s) = sA 0 - A where only A 0 is singular, or both A 0 and A are singular. Nonsingular systems are considered as a particular case of singular systems. The presented algorithm of the matrix partial fraction expansion is suitable to determine the matrix transfer function, and is computer oriented because all manipulations can be performed on matrices with constant entries only.
TL;DR: In this paper, a sufficient condition for strong shift equivalence for n × n matrices with n > 2 was presented, where the vertices of a directed graph are all n × N nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities.
TL;DR: In this paper, it was shown that in characteristic 0 all closures of nilpotent conjugacy classes have been proved to be normal and Cohen-Macaulay in arbitrary characteristic, thus generalizing a result of Hesselink [3] to arbitrary characteristic.
TL;DR: In this article, it was shown that any complex singular square matrix T is a product of two nilpotent matrices A and B with rank A = rank B = rank T except when T is an n X n complex matrix T = 0.
TL;DR: In this article, the authors present elementary proofs of the factorization of a square matrix into two hermitian or symmetric matrices, and prove that the factorisation can be expressed as
Abstract: This paper presents elementary proofs of the factorization of a square matrix into two hermitian or symmetric matrices.
Patent•
IBM1
TL;DR: In this paper, a vector generator for an all-points-addressable frame buffer capable of the non-word aligned access, simultaneously, of a square M by N array of pixels providing fast vector drawing independently of vector slope and position in the whole screen area of an attached display monitor.
Abstract: A vector generator for us with an all-points-addressable frame buffer capable of the non-word aligned access, simultaneously, of a square M by N array of pixels providing fast vector drawing independently of vector slope and position in the whole screen area of an attached display monitor. The vector generator utilizes a triangular logic matrix together with a line drawing unit to generate M vector bits lying in an M by N square matrix of the screen of an attached monitor in one memory cycle of the frame buffer and uses the generated matrix to generate a direct mask for the frame buffer whereby the M bit vector may be stored in a single memory cycle.
TL;DR: In this paper, the transfer function matrix of linear multivariable singular systems (i.e., systems of the form E\dot{x} =Ax + Bu, with E singular) is determined directly in terms of the matrices of the state-space description and the characteristic polynomial without inverting a polynomial matrix.
Abstract: A formula is given, which allows the determination of the transfer function matrix of linear multivariable singular systems (i.e., systems of the form E\dot{x} =Ax + Bu , with E singular) directly in terms of the matrices of the state-space description and the characteristic polynomial without inverting a polynomial matrix. An alternative closed-form expression for the transfer function matrix in terms of the pencil A - sE is also given.
TL;DR: In this paper, a constructive procedure for generating nonsingular solutions of the matrix equation XA = A T X and establish an interesting relationship between a given solution X of the above equation and the associated matrix polynomial p (A ).
TL;DR: It is given necessary and sufficient conditions for a distance matrix to have a unicyclic graph as unique optimal graph realization.
TL;DR: In this paper, it was shown that a nonsingular p-by-p matrix A is an inverse M -matrix if and only if Q T AQ + D is an n -by-n inverse M-matrix whenever Q is a p -by n nonnegative matrix with exactly one positive entry in each column and D is a positive diagonal matrix.
TL;DR: A classification for the triangular factorization of square matrices is given and the Gauss—Jordan elimination algorithm in a version which admits efficient implementation on a vector computer is described.
Posted Content•
01 Jan 1987TL;DR: In this paper, the authors consider the question of whether there exists an invertible lower triangular matrix L such that L-1AL is upper triangular, and if so, what can be said about the order in which the eigenvalues of A may appear on the diagonal of L 1AL.
Abstract: This paper is concerned with the following questions. Given a square matrix A, when does there exist an invertible lower triangular matrix L such that L-1AL is upper triangular ? And if so, what can be said about the order in which the eigenvalues of A may appear on the diagonal of t-1AL ? The motivation for considering these questions comes from systems theory. In fact they arise in the study of complete factorizations of rational matrix functions. There is also an intimate connection with the problem of complementary triangularization of pairs of matrices discussed in [4].