Showing papers on "Matrix analysis published in 2002"

Products of random matrices.

Abstract: We derive analytic expressions for infinite products of random 2 x 2 matrices. The determinant of the target matrix is log-normally distributed, whereas the remainder is a surprisingly complicated function of a parameter characterizing the norm of the matrix and a parameter characterizing its skewness. The distribution may have importance as an uncommitted prior in statistical image analysis.

Inverses of 2 × 2 block matrices

Abstract: In this paper, the authors give explicit inverse formulae for 2 × 2 block matrices with three different partitions. Then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as Hamiltonian, per-Hermitian, and centro-Hermitian matrices.

Data-sparse approximation by adaptive H 2 -matrices

Abstract: A class of matrices (H2-matrices) has recently been introduced for storing discretisations of elliptic problems and integral operators from the BEM. These matrices have the following properties: (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but after truncation to the H2-matrix format these operations are again of quasi-linear complexity.We introduce the basic ideas of H- and H2-matrices and present an algorithm that adaptively computes approximations of general matrices in the latter format.

An introduction to hierarchical matrices

Abstract: We give a short introduction to a method for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods or as the inverses of partial differential operators. The result of the approximation will be the so-called hierarchical matrices (or short $\mathcal {H}$-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix-matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity.

Scattering-matrix analysis of linear periodic arrays

Abstract: A spherical-wave source scattering-matrix description of acoustic radiators, along with reciprocity and power conservation, is applied to analyze infinite and finite linear periodic arrays that support traveling waves. We prove that for a general linear periodic array of small radiators, a traveling wave must be a slow wave with a propagation constant /spl beta/ greater than the propagation constant k of the medium in which the array is located. For an infinite periodic linear array of small isotropic radiators, the scattering-matrix analysis leads to a closed-form expression for the propagation constant of the traveling wave in terms of the normalized separation distance kd and the phase of the effective scattering coefficient of the array elements. These two parameters are the only critical variables in the N/spl times/N matrix equation, for N radiation coefficients, that is derived for a finite linear array of N elements. Resonances in the curves of total power radiated versus kd for a finite array excited with one feed element demonstrate the existence of the traveling wave predicted for the corresponding infinite array. The computed power curves, as well as directivity patterns, illustrate that the finite array becomes a more efficient endfire radiator as /spl beta/ approaches the value of k. The maximum attainable endfire directivity of a finite array with a single feed element is a monotonically increasing function of the phase velocity of the traveling wave, and this function is practically independent of the parameters of the array used to obtain this phase velocity. The basic formulation applies to any array composed of linear, reciprocal, lossless array elements, such as small linear periodic antennas.

Eigensolutions for matrices of special structures

Abstract: In this article, a simple and efficient method is developed for calculating the eigenvalues and eigen vectors of matrices having special structures. This is achieved by decomposing the matrices into specific forms. The application is extended to the eigensolution of the Laplacian matrices of symmetric graphs. Copyright © 2003 John Wiley & Sons, Ltd.

Concentration of norms and eigenvalues of random matrices

Abstract: We prove concentration results for $\ell_p^n$ operator norms of rectangular random matrices and eigenvalues of self-adjoint random matrices. The random matrices we consider have bounded entries which are independent, up to a possible self-adjointness constraint. Our results are based on an isoperimetric inequality for product spaces due to Talagrand.

A CONSTRUCTIVE METHOD FOR FINDING fl-INVARIANT MEASURES FOR TRANSITION MATRICES OF M=G=1 TYPE

Abstract: In this paper, we study the transition matrix of M=G=1 type. The radius of convergence is discussed, conditions on the fi-classiflcation of the states are obtained, and expressions of the fl-invariant measure are constructed. The censoring technique is generalized to deal with nonnegative matrices, which may be neither stochastic nor substochastic. This allows us to prove a factorization result for the discounted transition matrix. This factorization provides a unifled algorithmic approach for expressing the fl-invariant measure for transition matrices with a block-structure, including the matrix of M=G=1 type.

Mirrorsymmetric Matrices, Their Basic Properties, and an Application on Odd/Even-Mode Decomposition of Symmetric Multiconductor Transmission Lines

Abstract: Mirrorsymmetric matrices, which are the interaction matrices of mirrorsymmetric structures, are defined in this paper. The well-known centrosymmetric matrices, which can only reflect the mirror reflection relations of mirrorsymmetric structures with no component or one component on the mirror plane, are special cases of mirrorsymmetric matrices. However, almost all the properties of centrosymmetric matrices can be directly generalized to mirrorsymmetric matrices. It is proved that the eigenvectors of a mirrorsymmetric matrix are either mirrorsymmetric or skew-mirrorsymmetric corresponding to even-modes and odd-modes of the real physical systems. The application on odd/even-mode decomposition of symmetric multiconductor transmission lines is investigated in detail.

Linear algebra for skew-polynomial matrices

Abstract: We describe an algorithm for transforming skew-polynomial matrices over an Ore domain in row-reduced form, and show that this algorithm can be used to perform the standard calculations of linear algebra on such matrices (ranks, kernels, linear dependences, inhomogeneous solving). The main application of our algorithm is to desingularize recurrences and to compute the rational solutions of a large class of linear functional systems. It also turns out to be efficient when applied to ordinary commutative matrix polynomials.

A generalization of totally unimodular and network matrices

Abstract: In this thesis we discuss possible generalizations of totally unimodular and network matrices. Our purpose is to introduce new classes of matrices that preserve the advantageous properties of these well-known matrices. In particular, our focus is on the polyhedral consequences of totally unimodular matrices, namely we look for matrices that can ensure vertices that are scalable to an integral vector by an integer k. We argue that simply generalizing the determinantal structure of totally unimodular matrices does not suffice to achieve this goal and one has to extend the range of values the inverses of submatrices can contain. To this end, we define k-regular matrices. We show that k-regularity is a proper generalization of total unimodularity in polyhedral terms, as it guarantees the scalability of vertices. Moreover, we prove that the k-regularity of a matrix is necessary and sufficient for substituting mod-k cuts for rank-1 Chvatal-Gomory cuts. In the second part of the thesis we introduce binet matrices, an extension of network matrices to bidirected graphs. We provide an algorithm to calculate the columns of a binet matrix using the underlying graphical structure. Using this method, we prove some results about binet matrices and demonstrate that several interesting classes of matrices are binet. We show that binet matrices are 2-regular, therefore they provide half-integral vertices for a polyhedron with a binet constraint matrix and integral right hand side vector. We also prove that optimization on such a polyhedron can be carried out very efficiently, as there exists an extension of the network simplex method for binet matrices. Furthermore, the integer optimization with binet matrices is equivalent to solving a matching problem. We also describe the connection of k-regular and binet matrices to other parts of combinatorial optimization, notably to matroid theory and regular vectorspaces.

Perturbed Markov Chains

Abstract: We obtain results on the sensitivity of the invariant measure and other statistical quantities of a Markov chain with respect to perturbations of the transition matrix. We use graph-theoretic techniques, in contrast with the matrix analysis techniques previously used.

Cycles of Nonzero Elements in Low Rank Matrices

Abstract: We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices.

Basic Matrix Algebra with Algorithms and Applications

Abstract: SYSTEMS OF LINEAR EQUATIONS AND THEIR SOLUTION Recognizing Linear Systems and Solutions Matrices, Equivalence and Row Operations Echelon Forms and Gaussian Elimination Free Variables and General Solutions The Vector Form of the General Solution Geometric Vectors and Linear Functions Polynomial Interpolation MATRIX NUMBER SYSTEMS Complex Numbers Matrix Multiplication Auxiliary Matrices and Matrix Inverses Symmetric Projectors, Resolving Vectors Least Squares Approximation Changing Plane Coordinates The Fast Fourier Transform and the Euclidean Algorithm. DIAGONALIZABLE MATRICES Eigenvectors and Eigenvalues The Minimal Polynomial Algorithm Linear Recurrence Relations Properties of the Minimal Polynomial The Sequence {Ak} Discrete dynamical systems Matrix compression with components DETERMINANTS Area and Composition of Linear Functions Computing Determinants Fundamental Properties of Determinants Further Applications Appendix: The abstract setting Selected practice problem answers Index

A class of robustness problems in matrix analysis

Abstract: We present an overview of several results and a literature guide, prove some new results, and state open problems concerning description of all robust matrices in the following sense: Let be given a class of real or complex matrices A, and for each X ∈ A, a set G(X) is given. An element Y 0∈G(X 0) will be called robust (relative to the sets A and G(X) if for every X ∈A close enough to X 0 there is a X ∈ G(X) that is as close to Y 0 as we wish. The following topics are covered, with respect to the robustness property: 1. Invariant subspaces of matrices; here the set G(X) is the set of all X-invariant subspaces. 2. Invariant subspaces of matrices with symmetries related to indefinite inner products. The invariant subspaces in question include semidefinite and neutral subspaces (with respect to an indefinite inner product). 3. Applications of invariant subspaces of matrices with or without symmetries. The applications include: general matrix quadratic equations, the continuous and discrete algebraic Riccati equations, minimal factorization of rational matrix functions with symmetries and the transport equation from mathematical physics. 4. Several matrix decompositions: polar decompositions with respect to an indefinite inner product, Cholesky factorizations, singular value decomposition.

On Some Properties of Convex Matrix Sets Characterized by P -Matrices and Block P -Matrices

Abstract: In this paper, we consider convex sets of real matrices and establish criteria characterizing these sets with respect to certain matrix properties of their elements. In particular, we deal with convex sets of P-matrices, block P-matrices and M-matrices, nonsingular and full rank matrices, as well as stable and Schur stable matrices. Our results are essentially based on the notion of a block P-matrix and extend and generalize some recently published results on this topic.

Some inequalities for the Khatri-Rao product of matrices.

Abstract: Several inequalities for the Khatri-Rao product of complex positive definite Hermitian matrices are established, and these results generalize some known inequalities for the Hadamard and Khatri-Rao products of matrices.

Efficient linear algebra routines for symmetric matrices stored in packed form

Abstract: Quantum chemistry methods require various linear algebra routines for symmetric matrices, for example, diagonalization or Cholesky decomposition for positive matrices. We present a small set of these basic routines that are efficient and minimize memory requirements.

Introduction to matrix theory : with applications to business and economics

Abstract: Vectors and Matrices Vector Spaces and Inner-Product Spaces Systems of Linear Equations and Inverses of Matrices Determinants Linear Mappings and Matrices Eigenvalues, Invariant Subspaces, Canonical Forms Special Matrices Elements of Matrix Analysis.