Showing papers on "Square matrix published in 1996"

An algorithm for matrix extension and wavelet construction

Abstract: This paper gives a practical method of extending an n x r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z) has rank r for each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.

A J matrix engine for density functional theory calculations

Abstract: We introduce a new method for the formation of the J matrix (Coulomb interaction matrix) within a basis of Cartesian Gaussian functions, as needed in density functional theory and Hartree–Fock calculations. By summing the density matrix into the underlying Gaussian integral formulas, we have developed a J matrix ‘‘engine’’ which forms the exact J matrix without explicitly forming the full set of two electron integral intermediates. Several precomputable quantities have been identified, substantially reducing the number of floating point operations and memory accesses needed in a J matrix calculation. Initial timings indicate a speedup of greater than four times for the (pp‖pp) class of integrals with speedups increasing to over ten times for (ff‖ff ) integrals.

A Characterization and Representation of the Drazin Inverse

Abstract: We establish characterization and representation for the Drazin inverse of an arbitrary square matrix which reduce to the well-known result if the matrix is nonsingular.

An inequality for nonnegative matrices and the inverse eigenvalue problem

Abstract: We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases.

Quadratic eigenproblems are no problem

Abstract: High-dimensional eigenproblems often arise in the solution of scientific problems involving stability or wave modeling. In this article we present results for a quadratic eigenproblem that we encountered in solving an acoustics problem, specifically in modeling the propagation of waves in a room in which one wall was constructed of sound-absorbing material. Efficient algorithms are known for the standard linear eigenproblem, Ax = x where A is a real or complex-valued square matrix of order n. Generalized eigenproblems of the form Ax = Bx, which occur in nite element formulations, are usually reduced to the standard problem, in a form such as B Ax = x. The reduction requires an expensive inversion operation for one of the matrices involved. Higher-order polynomial eigenproblems are also usually transformed into standard eigenproblems. We discuss here the second-degree (i.e., quadratic) eigenproblem 2C2 + C1 + C0 x = 0 in which the matrices Ci are square matrices.

A Chain Rule for Matrix Functions and Applications

Abstract: Let $f$ be a not necessarily analytic function and let $A(t)$ be a family of $n \times n$ matrices depending on the parameter $t$. Conditions for the existence of the first and higher derivatives of $f(A(t))$ are presented together with formulae that represent these derivatives as a submatrix of $f(B)$, where $B$ is a larger block Toeplitz matrix. This block matrix representation of the first derivative is shown to be useful in the context of condition estimation for matrix functions. The results presented here are slightly stronger than those in the literature and are proved in a considerably simpler way.

Wiener and Hyper-Wiener Numbers in a Single Matrix

Abstract: A novel unsymmetric square matrix, CJu, is proposed for calculating both Wiener,1 W, and hyper-Wiener,2 WW, numbers. This matrix is constructed by using the principle of single endpoint characterization of paths.3 Its relation with Wiener-type numbers is discussed.

The generalized linear complementarity problem revisited

Abstract: Given a vertical block matrixA, we consider in this paper the generalized linear complementarity problem VLCP(q, A) introduced by Cottle and Dantzig. We formulate this problem as a linear complementarity problem with a square matrixM, a formulation which is different from a similar formulation given earlier by Lemke. Our formulation helps in extending many well-known results in linear complementarity to the generalized linear complementarity problem. We also show that the class of vertical block matrices which Cottle and Dantzig's algorithm can process is the same as the class of equivalent square matrices which Lemke's algorithm can process. We also present some degree-theoretic results on a vertical block matrix.

Best linear predictors for factor scores

Abstract: From the literature three types of predictors for factor scores are available These are characterized by the constraints: linear, linear conditionally unbiased, and linear correlation preserving Each of these constraints generates a class of predictors Best predictors are defined in terms of Lowner's partial matrix order applied to matrices of mean square error of prediction It is shown that within the first two classes a best predictor exists and that it does not exist in the third

State-space parametrizations of multivariable linear systems using tridiagonal matrix forms

Abstract: Tridiagonal parametrizations of linear state-space models are proposed for multivariable system identification. The parametrizations are surjective, i.e. all systems up to a given order can be described. The parametrization is based on the fact that any real square matrix is similar to a real tridiagonal form as well as a compact tridiagonal form. These parametrizations has significantly fewer parameters compared to a full parametrization of the state-space matrices.

Method and system for establishing a cryptographic key agreement using linear protocols

Abstract: A method for establishing key agreement between two communicating parties using a general linear protocol in finite and infinite dimensional spaces. Two topological linear spaces, in particular Euclidean spaces, and a non-trivial degenerate linear operator are selected. Each party respectively selects a secret element, and exchanges with the other party an image under the transformation of a matrix. Key agreement is therefore mutually established between the two communicating parties having the same cryptographic key. Various illustrative embodiments of the general linear operator are disclosed, including a rectangular matrix, a square matrix, a symmetric matrix, a skew symmetric matrix, an upper triangular square matrix, a lower triangular square matrix, a special type of skew symmetric matrix to generate a modified cross product protocol, a series of matrices to generate a sequential key protocol, and a combination of circulant matrices.

Dykstra's Algorithm for a Constrained Least-squares Matrix Problem

Abstract: We apply Dykstra's alternating projection algorithm to the constrained least-squares matrix problem that arises naturally in statistics and mathematical economics. In particular, we are concerned with the problem of finding the closest symmetric positive definite bounded and patterned matrix, in the Frobenius norm, to a given matrix. In this work, we state the problem as the minimization of a convex function over the intersection of a finite collection of closed and convex sets in the vector space of square matrices. We present iterative schemes that exploit the geometry of the problem, and for which we establish convergence to the unique solution. Finally, we present preliminary numberical results to illustrate the performance of the proposed iterative methods.

Surface multiple attenuation via eigenvalue decomposition

Abstract: A marine seismic signal is transformed from time domain into frequency domain and represented by matrix D. The marine data signal is truncated in time, transformed into the frequency domain and represented by matrix D T . Eigenvalue decomposition D T =S·Λ·S -1 of matrix D T is computed. Matrix product D·S is computed and saved in memory. Matrix inverse S -1 is computed and saved in memory. An initial estimate of the source wavelet w is made. Source wavelet w is computed by iterating the steps of computing diagonal matrix [I-w -1 Λ], computing matrix inverse [I-w -1 Λ] -1 , retrieving matrix product D·S and matrix inverse S -1 from memory, and minimizing the total energy in matrix product [D·S] [I-w -1 Λ] -1 S -1 . Primary matrix P representing the wavefield free of surface multiples is computed by inserting computed value for w into the expression [D·S] [I-w -1 Λ] -1 S -1 . Primary matrix P is inverse transformed from frequency domain into time domain.

Supersymmetric Generalizations of Matrix Models

Abstract: In this thesis generalizations of matrix and eigenvalue models involving supersymmetry are discussed. Following a brief review of the Hermitian one matrix model, the c=-2 matrix model is considered. Built from a matrix valued superfield this model displays supersymmetry on the matrix level. We stress the emergence of a Nicolai-map of this model to a free Hermitian matrix model and study its diagrammatic expansion in detail. Correlation functions for quartic potentials on arbitrary genus are computed, reproducing the string susceptibility of c=-2 Liouville theory in the scaling limit. The results may be used to perform a counting of supersymmetric graphs. We then turn to the supereigenvalue model, today's only successful discrete approach to 2d quantum supergravity. The model is constructed in a superconformal field theory formulation by imposing the super-Virasoro constraints. The complete solution of the model is given in the moment description, allowing the calculation of the free energy and the multi-loop correlators on arbitrary genus and for general potentials. The solution is presented in the discrete case and in the double scaling limit. Explicit results up to genus two are stated. Finally the supersymmetric generalization of the external field problem is addressed. We state the discrete super-Miwa transformations of the supereigenvalue model on the eigenvalue and matrix level. Properties of external supereigenvalue models are discussed, although the model corresponding to the ordinary supereigenvalue model could not be identified so far.

Matrix inequalities involving a positive linear map

Abstract: Let A be a Hermitian matrix, let Φ be a normalized positive linear map and let f be a continuous real valued function. Real constants α and β such that are determined. If f is matrix convex then β can be taken to be 1. A unified approach is proposed so that the problem of determining α and β is reduced to solving a single variable convex minimization problem. As an illustration, the results are applied to the power functions.

A new 2 × 2 matrix representation for twisted nematic liquid crystal displays at oblique incidence

Abstract: A new 2×2 matrix representation for the twisted nematic liquid crystal display (TN-LCD) at oblique incidence was obtained. Compared with the previous representation developed by Lien [Appl. Phys. Lett. 57 (1990) 2767 and SID 91 Dig. (1991) 586], the optical transmissions calculated by this new 2×2 matrix method are much closer to those calculated by the 4×4 matrix method with spectrum averaging to account for the nonzero band-width of the incident light. The discrepancy between the last two is generally less than 1%. The simplicity and accuracy makes this new 2×2 matrix method very useful.

A fast block Hankel solver based on an inversion formula for block Loewner matrices

Abstract: We propose a newO(p 3 n 2) algorithm for solving complexnp×np linear systems that have block Hankel structure, where the blocks are square matrices of sizep×p. Via FFTs the block Hankel system is transformed into a block Loewner system. An inversion formula enables us to calculate the inverse of the block Loewner matrix explicitely. The parameters that occur in this inversion formula are calculated by solving two rational interpolation problems on the unit circle. We have implemented our algorithm in Fortran 90. Numerical examples are included.

Conditions for the positivity of determinants

Abstract: Consider a matrix with positive diagonal entries, which is similar via a positive diagonal matrix to a symmetric matrix, and whose signed directed graph has the property that if a cycle and its symmetrically placed complement have the same sign, then they are both positive. We provide sufficient conditions so that A be a P-matrix, that is , a matrix whose principal minors are all positive. We further provide sufficiet conditions for an arbitrary matrix A whose (undirected) graph is subordinate to a tree, to be a P-matrix. If, in additionA is sign symmetric and its undirected graph is a tree, we obtain necessary and sufficient conditions that it be a P-matrix. We go on to consider the positive semi-definiteness of symmetric matrices whose graphs are subordinate to a given tree and discuss the convexity of the set of all such matrices.

The recurrent algorithm of the rigorous solving 1-dimensional wave equations in multilayered media

Abstract: The rigorous difference equations for the elements of any order of transfer matrices of coupled waves and the renormalization relations for their recursion coefficients under homogeneous and inhomogeneous rescaling are obtained. Incidentally, the generalization of Abeles theorem to arbitrary-dimensional square matrix formulae are deduced.

Idempotent factorization of matrices

Abstract: Necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k ≥ 1 idempotent matrices of prescribed nullities.