Certain Class Of Matrices Research Articles

New polynomial-time algorithms for Camion bases

Let M be a finite set of vectors in R n of cardinality m and H ( M ) = { { x ∈ R n : c T x = 0 } : c ∈ M } the central hyperplane arrangement represented by M. An independent subset of M of cardinality n is called a Camion basis, if it determines a simplex region in the arrangement H ( M ) . In this paper, we first present a new characterization of Camion bases, in the case where M is the column set of the node-edge incidence matrix (without one row) of a given connected digraph. Then, a general characterization of Camion bases as well as a recognition procedure which runs in O ( n 2 m ) are given. Finally, an algorithm which finds a Camion basis is presented. For certain classes of matrices, including totally unimodular matrices, it is proven to run in polynomial time and faster than the algorithm due to Fonlupt and Raco.

Basic Properties of the (t, n)-Threshold Visual Secret Sharing Scheme with Perfect Reconstruction of Black Pixels

In this paper we consider the (t, n)-threshold visual secret sharing scheme (VSSS) in which black pixels in a secret black-white images is reproduced perfectly as black pixels when we stack arbitrary t shares. This paper provides a new characterization of the (t, n)-threshold visual secret sharing scheme with such a property (hereafter, we call such a VSSS the (t, n)-PBVSSS for short). We use an algebraic method to characterize basis matrices of the (t, n)-PBVSSS in a certain class of matrices. We show that the set of all homogeneous polynomials each element of which yields basis matrices of the (t, n)-PBVSSS becomes a set of lattice points in an (n?t+1)-dimensional linear space. In addition, we prove that the optimal basis matrices in the sense of maximizing the relative difference among all the basis matrices in the class coincides with the basis matrices given by Blundo, Bonis and De Santis [3] for all n? t ? 2.

Further properties of the star, left-star, right-star, and minus partial orderings

Certain classes of matrices are indicated for which the star, left-star, right-star, and minus partial orderings, or some of them, are equivalent. Characterizations of the left-star and right-star orderings, similar to those devised by Hartwig and Styan [Linear Algebra Appl. 82 (1986) 145] for the star and minus orderings, are established along with other auxiliary results, which are of independent interest as well. Some inheritance-type properties of matrices are also given. The class of EP matrices appears to be essential in several points of our considerations.

Existence and construction of nonnegative matrices with complex spectrum

The following inverse spectrum problem for nonnegative matrices is considered: given a set of complex numbers σ={ λ 1, λ 2,…, λ n }, find necessary and sufficient conditions for the existence of an n× n nonnegative matrix A with spectrum σ. Our work is motivated by a relevant theoretical result of Guo Wuwen [Linear Algebra Appl. 266 (1997) 261, Theorem 2.1]: there exists a real parameter λ 0⩾max 2⩽ j⩽ n | λ j | such that the problem has a solution if and only if λ 1⩾ λ 0. In particular, we discuss how to compute λ 0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.

Conservatism of randomized structured singular value

In this note, we study the conservatism of structured singular value computation by a randomized algorithm. It is proved that, if the maximization problem μ(M) = max ρ(MΔ) is solved by generating polynomial number of random Δ samples and then taking the maximum of the function at these sample points, for any fixed lower bound on the confidence level, conservatism of the resulting estimate grows faster than any polynomial function of the logarithm of the matrix size. This result holds for purely complex, mixed, and purely real cases with no repeated blocks. However, it is shown to be not true if the number of samples exceeds some exponential function for the purely complex version of the problem. Although the estimate obtained by polynomial number of samples can be used to find an exact robustness margin with a high confidence level, for all except for a small relative volume of the uncertainty set, it has high conservatism for the worst-case robustness analysis, no matter how small the confidence level lower bound may be. The results of this note imply that conservatism will be large for certain classes of matrices. However, this does not eliminate the possibility of existence of other classes of matrices for which the conservatism is small.

Analytic inversion for a general case of certain classes of matrices

The explicit inverse and determinant of a class of matrices are presented. The class under consideration is defined by 4n − 2 parameters, analytic expressions of which form the elements of the upper Hessenberg type inverse. These analytic expressions enable recursion formulae to be obtained, which reduce the arithmetic operations to O(n 2). The Hadamard product of two specific structures' matrices provides the class presented, special cases of which are already known classes of test matrices.

Spectral criteria for complementary triangular forms

In this paper we consider the following problem: Given two matricesA,Z∈ℂ n×n , does there exist an invertiblen×n-matrixS such thatS−1AS is an upper triangular matrix andS −1 ZS is a lower triangular matrix, and if so, what can be said about the order in which the eigenvalues ofA andZ appear on the diagonals of these triangular matrices? For special choices ofA andZ a complete solution is possible, as has been shown by several authors. Here we follow a lead, provided by Shmuel Friedland, who discussed the case where bothA andZ have at leastn-1 linearly independent eigenvectors, and we descibe the problem in terms of Jordan chains and left-Jordan chains for the matricesA, Z. The results give some insight in the question why certain classes of matrices (like the nonderogatory and the rank 1 matrices) allow for a detailed solution of the problems described above; for some of these classes the result of this analysis is presented here for the first time.

A Minimax Theorem and a Dulmage–Mendelsohn Type Decomposition for a Class of Generic Partitioned Matrices

This paper discusses an extension of the Dulmage--Mendelsohn decomposition for a certain class of matrices whose row-set and column-set are divided into couples or singletons. A genericity assumption is imposed and an admissible transformation is defined in respect of this partition structure. Extensions of the Konig--Egervary theorem and the Hall--Ore theorem are established. The latter states that the rank of such a matrix is characterized by the minimum value of a submodular function, of which the set of minimizers yields a canonical block-triangularization under the admissible transformations.

A sufficient condition for stability of a polytope of matrices

In this paper, we study the stability of interval dynamical systems. A sufficient condition for the stability of a polytope of matrices, which is shown to be necessary and sufficient for a certain class of matrices, is obtained. The results developed can also be applied to the stability of a positive cone of matrices and sufficient conditions for the stability of interval dynamical systems are obtained. A relationship between real parts of eigenvalues and matrix measures is also presented.

Orderings, Multicoloring, and Consistently Ordered Matrices

The use of multicoloring as a means for the efficient implementation of diverse iterative methods for the solution of linear systems of equations, arising from the finite difference discretization of partial differential equations, on both parallel (concurrent) and vector computers has been extensive; these include SOR-type and preconditioned conjugate gradient methods as well as smoothing procedures for use in multigrid methods. Multicolor orderings, corresponding to reorderings of the points of the discretization, often allow a local decoupling of the unknowns. Some new theory is presented which allows one to quickly verify whether or not a member of a certain class of matrices is consistently ordered (or $\pi $-consistently ordered) solely by looking at the structure of the matrix under consideration. This theory allows one to quickly ascertain that, while many well-known multicoloring schemes do give rise to coefficient matrices which are consistently ordered, many others do not. Some alternative orderings and multicoloring schemes proposed in the literature are surveyed and the theory is applied to the resulting coefficient matrices.

Optimum first and second order extrapolations of Successive Overrelaxation type methods for certain classes of matrices

This paper deals with the iterative solution of the linear systemx=Bx+c when its Jacobi matrixB is weakly 2-cyclic consistently ordered and has a complex eigenvalue spectrum which lies on a straight-line segment. The optimization problem of the following three methods is considered and solved: i) The extrapolation of the optimum Successive Overrelaxation (SOR) ii) The second order extrapolation of a “good” SOR and iii) The second order extrapolation of the Gauss-Seidel method. In addition a variant of the second order methods considered, suitable for the solution of the system even ifB isnot necessarily weakly 2-cyclic consistently ordered, is proposed. Finally a reference to a theoretical comparison of the various optimum methods in the paper is made and their asymptotic convergence factors for selected eigenvalue spectra are illustrated in a Table in support of the theory developed.

Doubly Lexical Orderings of Matrices

Every matrix has a doubly lexical ordering an ordering of the rows and columns so that the row vectors are lexically (or “lexicographically”) increasing and the column vectors are lexically increasing. Every graph has a lexical ordering: a vertex ordering making the neighbourhood matrix doubly lexical. An almost linear time doubly lexical ordering algorithm is given. Doubly lexical orderings unify the orderings characterizing certain classes of matrices and graphs, including totally balanced matrices, subtree matrices and chordal graphs.

Eigenvalue interlacing for certain classes of matrices with real principal minors

Two common properties of Z-matrices and Hermitian matrices are considered: (1) The eigenvalue interlacing property, i.e., the two smallest real eigenvalues of a matrix are interlaced by the smallest real eigenvalue of any principal submatrix of order one less. (2) The positive GLP property, i.e., if a matrix has a positive sequence of generalized leading principal minors, then all the principal minors of the matrix are positive. The relationship between these properties as well as related properties is examined in general.

More results on the convergence of iterative methods for the symmetric linear complementarity problem

In an earlier paper, the author has given some necessary and sufficient conditions for the convergence of iterative methods for solving the linear complementarity problem. These conditions may be viewed as global in the sense that they apply to the methods regardless of the constant vector in the linear complementarity problem. More precisely, the conditions characterize a certain class of matrices for which the iterative methods will converge, in a certain sense, to a solution of the linear complementarity problem for all constant vectors. In this paper, we improve on our previous results and establish necessary and sufficient conditions for the convergence of iterative methods for solving each individual linear complementarity problem with a fixed constant vector. Unlike the earlier paper, our present analysis applies only to the symmetric linear complementarity problem. Various applications to a strictly convex quadratic program are also given.

Limit probabilities for coranks of matrices over GF(q)

In this paper we consider certain classes of matrices over the finite fields GF(q): namely arbitrary s × t matrices with s  t, symmetric s × s matrices, skew symmetric s × s matrices when q is odd, and antisymmetric s × s matrices when q = 2. For a matrix M of one of these types, we define corank M=s- rank M. Our main results give limit probabilitics as for coranks of these types of mairices.

Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues

The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA1, A2, …, Ap, is the least integert for wich there existt diadsu(r)v(r)τ, τ=1,2,...,t, such that\(A_s = \mathop \sum \limits_{r = 1}^t w_{s,r} u^{(r)} v^{(r)T} \). IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log2k) steps with max(n2+p (n+k), k2n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log2k) parallel steps and by a sequential algorithm inn(1+k2)+p (n+k)+0 (k3) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown.

On the mappingA→A+

In this note we discuss the invariance of certain classes of matrices under the mapping A→A + which replaces every entry of a complex matrix A by its absolute value.

The convergence of matrix transforms for certain Markov chains

In this paper it is shown that, for certain classes of matrices, the matrix transform of a periodic strongly ergodic stochastic matrix P converges.

On an integral representation of the function Tr(exp(A- B))

The conjecture that Tr(exp(A- lambda B)) can be written as a Laplace transform with a positive measure is proved for a certain class of matrices A and B. A few remarks are made about the undecided cases.

On k∞and other criticality conditions for an infinite system

Abstract It is shown that a study of certain class of matrices lead to criticality conditions based on the multigroup neutron transport model for a homogeneous infinite system. A consistent definition of k∞ is formulated. A simple example is presented.