Showing papers on "Square matrix published in 2006"

Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs

Abstract: We consider robust semi-definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description. On the basis of matrix sum-of-squares decompositions, we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart. With a novel matrix-version of Putinar's sum-of-squares representation for positive polynomials on compact semi-algebraic sets, we prove asymptotic exactness of the relaxation family under a suitable constraint qualification. If the uncertainty region is a compact polytope, we provide a new duality proof for the validity of Putinar's constraint qualification with an a priori degree bound on the polynomial certificates. Finally, we point out the consequences of our results for constructing relaxations based on the so-called full-block S-procedure, which allows to apply recently developed tests in order to computationally verify the exactness of possibly small-sized relaxations.

Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices

Abstract: Let A be an arbitrary matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has a large condition number. Using this result, we prove that it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we show that the smoothed precision necessary to solve Ax = b, for any b, using Gaussian elimination without pivoting is logarithmic. Moreover, when A is an all-zero square matrix, our results significantly improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 18 (1997), pp. 499-517).

Enumeration of symmetry classes of alternating sign matrices and characters of classical groups

Abstract: An alternating sign matrix is a square matrix with entries 1, 0 and ?1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinant and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs).

Eigenvalue Inequalities for Matrix Product

Abstract: We present a family of eigenvalue inequalities for the product of a Hermitian matrix and a positive-semidefinite matrix. Our theorem contains or extends some existing results on trace and eigenvalues

Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

Abstract: The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F-norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above-mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd.

A strong ramp secret sharing scheme using matrix projection

Abstract: This paper presents a strong (k,n) threshold-based ramp secret sharing scheme with k access levels The secrets are the elements represented in a square matrix S The secret matrix S can be shared among n different participants using a matrix projection technique where: i) any subset of k participants can collaborate together to reconstruct the secret, and ii) any subset of (k-1) or fewer participants cannot partially discover the secret matrix The primary advantages are its large compression rate on the size of the shares and its strong protection of the secrets

Solution to the Second-Order Sylvester Matrix Equation

Abstract: This note provides a complete general parametric solution ( ) to the generalized second-order Sylvester matrix equation + + = , with being an arbitrary square matrix. The primary feature of this solution is that the matrix does not need to be in any canonical form, or may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems involving second-order dynamical systems.

Solution to the second-order Sylvester matrix equation MVF/sup 2/+DVF+KV=BW

Abstract: This note provides a complete general parametric solution (V,W) to the generalized second-order Sylvester matrix equation MVF/sup 2/+DVF+KV=BW, with F being an arbitrary square matrix. The primary feature of this solution is that the matrix F does not need to be in any canonical form, or may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems involving second-order dynamical systems.

Termination of string rewriting with matrix interpretations

Abstract: A rewriting system can be shown terminating by an order-preserving mapping into a well-founded domain. We present an instance of this scheme for string rewriting where the domain is a set of square matrices of natural numbers, equipped with a well-founded ordering that is not total. The coefficients of the matrices can be found via a transformation to a boolean satisfiability problem. The matrix method also supports relative termination, thus it fits with the dependency pair method as well. Our implementation is able to automatically solve hard termination problems.

Smoothing-Norm Preconditioning for Regularizing Minimum-Residual Methods

Abstract: When GMRES (or a similar minimum-residual algorithm such as RRGMRES, MINRES, or MR-II) is applied to a discrete ill-posed problem with a square matrix, in some cases the iterates can be considered as regularized solutions. We show how to precondition these methods in such a way that the iterations take into account a smoothing norm for the solution. This technique is well established for CGLS, but it does not immediately carry over to minimum-residual methods when the smoothing norm is a seminorm or a Sobolev norm. We develop a new technique which works for any smoothing norm of the form $\|L\,x\|_2$ and which preserves symmetry if the coefficient matrix is symmetric. We also discuss the efficient implementation of our preconditioning technique, and we demonstrate its performance with numerical examples in one and two dimensions.

Method and device for coupling cancellation of closely spaced antennas

Abstract: The present invention relates to an antenna system (15) comprising at least two antenna radiating elements (RE1, RE2,... REN) and respective reference ports (R1, R2,... RN)1 the ports being defined by a symmetrical antenna scattering NxN matrix (S). The system (15) further comprises a compensating network (11 ) connected to the reference ports (R1, R2,... RN). The compensating network (11 ) is arranged for counteracting coupling between the antenna radiating elements (A1, A2... AN). The compensating network (11 ) is defined by a symmetrical compensating scattering 2Nx2N matrix (Sc) comprising four NxN blocks, the two blocks on the main diagonal containing all zeros and the other two blocks of the other diagonal containing a unitary NxN matrix (V) and its transpose (Vt). The product between the unitary matrix (V), the scattering NxN matrix (S) and the transpose (Vt) of the unitary matrix (V) equals an NxN matrix (s) which essentially is a diagonal matrix. The present invention also relates to a method for calculating a compensating scattering 2Nx2N matrix (Sc) for a compensating network (11 ) for an antenna system according to the above, and also to a compensating network (11 ) for an antenna system according to the above.

The reflexive solutions of the matrix equation AX B = C

Abstract: In this paper, we study the existence of a reflexive, with respect to the generalized reflection matrix P, solution of the matrix equation AX B = C. For the special case when B = I, we get the result of Peng and Hu [1].

Efficient filter weight computation for a mimo system

Abstract: Techniques to efficiently derive a spatial filter matrix are described. In a first scheme, a Hermitian matrix is iteratively derived based on a channel response matrix, and a matrix inversion is indirectly calculated by deriving the Hermitian matrix iteratively. The spatial filter matrix is derived based on the Hermitian matrix and the channel response matrix. In a second scheme, multiple rotations are performed to iteratively obtain first and second matrices for a pseudo-inverse matrix of the channel response matrix. The spatial filter matrix is derived based on the first and second matrices. In a third scheme, a matrix is formed based on the channel response matrix and decomposed to obtain a unitary matrix and a diagonal matrix. The spatial filter matrix is derived based on the unitary matrix, the diagonal matrix, and the channel response matrix.

Inverse scattering method for square matrix nonlinear Schr\"odinger equation under nonvanishing boundary conditions

Abstract: Matrix generalization of the inverse scattering method is developed to solve the multicomponent nonlinear Schr\"odinger equation with nonvanishing boundary conditions. It is shown that the initial value problem can be solved exactly. The multi-soliton solution is obtained from the Gel'fand--Levitan--Marchenko equation.

A Note on Weak and Strong Linearizations of Regular Matrix Polynomials

Abstract: A paper of Tan and Pugh [TP] raises the question of ambiguity in a frequently used form of linearization when applied to regular matrix polynomials. Here, further insight into this question is provided, as well as a reminder of a stronger form of linearization (for which ambiguities are removed) introduced by Gohberg et al. [GKL]. Let A0, A1, . . . , An ∈ Cn×n, and define the matrix polynomial L(λ) = ∑l i=0 λ Ai. Then L(λ) is said to be regular if det L(λ) is not identically equal to zero. We consider only regular matrix polynomials, and notice that Al = 0 is admitted. The degree of L(λ) is the largest j for which Aj 6= 0. Thus, it may happen that l > deg(L). Some important ideas for this discussion are as follows: Two regular matrix polynomials A(λ), B(λ) of the same size are said to be equivalent if there are unimodular matrix polynomials E(λ), F (λ) such that A(λ) = E(λ)B(λ)F (λ). The canonical form under equivalence is the well-known Smith form, and it reveals the structure of the invariant polynomials and (finite) elementary divisors. There is also a local Smith form in which the transforming matrices E(λ) and F (λ) are invertible near an eigenvalue λ0 and the elementary divisor structure of the single eigenvalue λ0 is revealed (see [BGR], for example). i.e. invertible with non-vanishing determinant independent of λ

Method and apparatus for singular value decomposition of a channel matrix

Abstract: A method and apparatus for decomposing a channel matrix in a wireless communication system are disclosed. A channel matrix H is generated for channels between transmit antennas and receive antennas (202). A Heπnitian matrix A=HHH or A=HHn is created (204). A Jacobi process is cyclically performed on the matrix A to obtain Q and DA matrixes such that A=QDAQH (206). DA is a diagonal matrix obtained by singular value decomposition (SVD) on the A matrix. In each Jacobi transformation, real part diagonalization is performed to annihilate real parts of off-diagonal elements of the matrix and imaginary part diagonalization is performed to annihilate imaginary parts of off -diagonal elements of the matrix after the real part diagonalization. U, V and DH matrixes of H matrix are then calculated from the Q and DA matrices. DH is a diagonal matrix comprising singular values of the H matrix (208).

Preserving geometric properties of the exponential matrix by block Krylov subspace methods

Abstract: Given a large square real matrix A and a rectangular tall matrix Q, many application problems require the approximation of the operation $\exp(A)Q$ . Under certain hypotheses on A, the matrix $\exp(A)Q$ preserves the orthogonality characteristics of Q; this property is particularly attractive when the associated application problem requires some geometric constraints to be satisfied. For small size problems numerical methods have been devised to approximate $\exp(A)Q$ while maintaining the structure properties. On the other hand, no algorithm for large A has been derived with similar preservation properties. In this paper we show that an appropriate use of the block Lanczos method allows one to obtain a structure preserving approximation to $\exp(A)Q$ when A is skew-symmetric or skew-symmetric and Hamiltonian. Moreover, for A Hamiltonian we derive a new variant of the block Lanczos method that again preserves the geometric properties of the exact scheme. Numerical results are reported to support our theoretical findings, with particular attention to the numerical solution of linear dynamical systems by means of structure preserving integrators.

Moments of minors of Wishart matrices

Abstract: For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order $m$ is populated by all $m\times m$-minors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by the fact that such minors arise in the expression of constraints on the covariance matrix in many classical multivariate problems.

On the complexity of some birational transformations

Abstract: Using three different approaches, we analyse the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach comprises the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis and the third method is more numerical, using integer arithmetics. These three methods have their own domain of application, but they give corroborating results, and lead us to a conjecture on the algebraic entropy of a class of maps constructed from matrix inversions.