Showing papers on "Matrix analysis published in 2009"
TL;DR: In this article, a statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices.
Abstract: The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes Anisotropy is discussed.
379 citations
TL;DR: Second-order Poincare inequalities (SOPE inequalities) as discussed by the authors were introduced to derive gaussian central limit theorems for Gaussian Toeplitz matrices.
Abstract: Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems The proofs of such results are usually rather difficult, involving hard computations specific to the model in question In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments In the process, we introduce a notion of ‘second order Poincare inequalities’: just as ordinary Poincare inequalities give variance bounds, second order Poincare inequalities give central limit theorems The proof of the main result employs Stein’s method of normal approximation A number of examples are worked out, some of which are new One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices
203 citations
TL;DR: In this article, the authors place themselves in the setting of high-dimensional statistical inference, where the number of variables $p$ in a data set of interest is of the same order of magnitude as the total number of observations $n$ for general population covariance, and derive a Mar\u{c}enko-Pastur-type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions.
Abstract: We place ourselves in the setting of high-dimensional statistical inference, where the number of variables $p$ in a data set of interest is of the same order of magnitude as the number of observations $n$. More formally, we study the asymptotic properties of correlation and covariance matrices, in the setting where $p/n\to\rho\in(0,\infty),$ for general population covariance. We show that, for a large class of models studied in random matrix theory, spectral properties of large-dimensional correlation matrices are similar to those of large-dimensional covarance matrices. We also derive a Mar\u{c}enko--Pastur-type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.
93 citations
Book•
02 Jul 2009
TL;DR: This self-contained textbook presents matrix analysis in the context of numerical computation with numerical conditioning of problems and numerical stability of algorithms at the forefront, using a unique combination of numerical insight and mathematical rigor to advance readers understanding of two phenomena.
Abstract: This self-contained textbook presents matrix analysis in the context of numerical computation with numerical conditioning of problems and numerical stability of algorithms at the forefront. Using a unique combination of numerical insight and mathematical rigor, it advances readers understanding of two phenomena: sensitivity of linear systems and least squares problems, and numerical stability of algorithms. This book differs in several ways from other numerical linear algebra texts. It offers a systematic development of numerical conditioning; a simplified concept of numerical stability in exact arithmetic; simple derivations; a high-level view of algorithms; and results for complex matrices. The material is presented at a basic level, emphasizing ideas and intuition, and each chapter offers simple exercises for use in the classroom and more challenging exercises for student practice. Audience: This book is intended for first-year graduate students in engineering, operations research, computational science, and all areas of mathematics. It also is appropriate for self-study. Contents: Preface; Introduction; Chapter 1: Matrices; Chapter 2: Sensitivity, Errors, and Norms; Chapter 3: Linear Systems; Chapter 4: Singular Value Decomposition; Chapter 5: Least Square Problems; Chapter 6: Subspaces; Index.
66 citations
TL;DR: Operational matrices of integration and product based on Chebyshev wavelets are presented and a general procedure for forming these matrices is given.
Abstract: Operational matrices of integration and product based on Chebyshev wavelets are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. Numerical examples are given to demonstrate applicability of these matrices.
49 citations
Journal Article•
TL;DR: This paper defines and illustrates sparse matrices, discusses the types of sparseMatrices, and provides examples of general data structures and algorithms for efficient solution of sparse matrix problems.
Abstract: Most college undergraduate computer science and mathematics textbooks on algorithms, matrix algebra, linear algebra, discrete math, or data structures do not provide even introductory material on the subject of sparse matrices. However in real world practice, most large and important applications of matrix algebra involve sparse matrices. Thus the computer science or CIS/IT student is ill prepared for real world applications in this field. This paper defines and illustrates sparse matrices, discusses the types of sparse matrices, and provides examples of general data structures and algorithms for efficient solution of sparse matrix problems.
47 citations
TL;DR: In this paper, Hartwig and Spindelbock demonstrate that Corollary 6 in [R.E. Hartwig, K. Köpcke] provides a powerful tool to investigate square matrices with complex entries.
43 citations
TL;DR: In this paper, the vibration and wave propagation in members of a framed structure are analyzed by the recently developed method of reverberation-ray matrix, based on one-dimensional theory of elastodynamics.
Abstract: Based on one-dimensional theory of elastodynamics, the vibration and wave propagation in members of a framed structure are analyzed by the recently developed method of reverberation-ray matrix. Unidirectional traveling wave solutions for axial, torsional, and two flexural waves in six modes that are reverberated in structural members through repeated reflection and multiple scattering by joints of the structure are expressed in matrix form with two sets of unknown amplitude coefficients. From joint coupling equations and from compatibility conditions of displacements in dual coordinates of each member, the two sets of unknowns are determined in terms of a reverberation-ray matrix for given source excitations at joints. Free and forced wave motion in steady-state response can all be evaluated from the matrix solutions, and transient response is determined in one more step of Fourier inverse transform. The method is particularly effective to determine the early time transient response through Neumann series expansion of the inverse transformed solution with a special numerical algorithm.
36 citations
TL;DR: The key result is to formally prove that the more distinct the r vectors of the r-column matrices are, the less they are swayed by noise.
Abstract: The task of finding a low-rank (r) matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering. The problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements. The former problem can be solved by concatenating a set of r-column matrices that share a common single r-dimensional solution space. Unfortunately, the number of possible submatrices is generally very large and, hence, the results obtained with one set of r-column matrices will generally be different from that captured by a different set. Ideally, we would like to find that solution that is least affected by noise. This requires that we determine which of the r-column matrices (i.e., which of the original feature points) are less influenced by the unknown noise term. This paper presents a criterion to successfully carry out such a selection. Our key result is to formally prove that the more distinct the r vectors of the r-column matrices are, the less they are swayed by noise. This key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r-column matrices. It is shown how this criterion can be effectively used to recover the noise-free matrix of rank r. Finally, we derive the affine and projective structure-from-motion (SFM) algorithms using the proposed criterion. Extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art.
32 citations
TL;DR: Some closed-form solutions are provided for the nonhomogeneous Yakubovich-conjugate matrix equation [email protected]?F=BY+R with X and Y being unknown matrices.
31 citations
30 Sep 2009
TL;DR: In this paper, a necessary condition for an underdeterministic linear system to have a unique nonnegative solution is derived, which requires that the measurement matrix have a row-span intersecting the positive orthant.
Abstract: This paper investigates conditions for an underdeter-mined linear system to have a unique nonnegative solution. A necessary condition is derived which requires the measurement matrix to have a row-span intersecting the positive orthant. For systems that satisfy this necessary condition, we provide equivalent characterizations for having a unique nonnegative solution. These conditions generalize existing ones to the cases where the measurement matrix may have different column sums. Focusing on binary measurement matrices especially ones that are adjacency matrices of expander graphs, we obtain an explicit threshold. Any nonnegative solution that is sparser than the threshold is the unique nonnegative solution. Compared with previous ones, this result is not only more general as it does not require constant degree condition, but also stronger as the threshold is larger even for cases with constant degree.
TL;DR: Random-matrix theory is applied to transition-rate matrices in the Pauli master equation to study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems.
Abstract: Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of independent rates of forward and backward transitions are considered. The first case leads to symmetric transition-rate matrices, whereas the second corresponds to general asymmetric matrices. The resulting matrix ensembles are different from the standard ensembles and show different eigenvalue distributions. For example, the fraction of real eigenvalues scales anomalously with matrix dimension in the asymmetric case.
TL;DR: In this paper, the authors studied the structure of the Fisher-Hartwig matrices with real α and β and 0 <α <|β| < 1, where the behavior is particularly simple.
Abstract: A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, N.T hey are parameterized by two constants, α and β. Their spectrum of eigenvalues has a simple asymptotic form in the limit as N goes to infinity. Here we study the structure of their eigenvalues and eigenvectors in this limiting case. We specialize to the case with real α and β and 0 <α <|β| < 1, where the behavior is particularly simple. The eigenvalues are labeled by an index l which varies from 0 to N − 1. An asymptotic analysis using Wiener-Hopf methods indicates that for large N ,t hejth component of the lth eigenvector varies roughly in the fashion ln ψ l ≈ ip l j +O (1/N ). The lth wavevector, p l ,v aries as
TL;DR: The limiting spectral distribution of a randomcirculant matrix is shown to be complex normal, and bounds are given for the probability that a circulant sign matrix is singular.
Abstract: This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random circulant matrix is shown to be complex normal, and bounds are given for the probability that a circulant sign matrix is singular.
TL;DR: In this article, the authors consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source and find the genus-zero algebraic function describing the limit mean density of eigenvalues.
Abstract: We consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source. We find the genus-zero algebraic function describing the limit mean density of eigenvalues in the case of an anharmonic potential and a diagonal external source with two symmetric eigenvalues. We discuss critical regimes where the density support changes the connectivity or increases the genus of the algebraic function and consequently obtain local universal asymptotic representations for the density at interior and boundary points of its support (in the generic cases). The investigation technique is based on an analysis of the asymptotic properties of multiple orthogonal polynomials, equilibrium problems for vector potentials with interaction matrices and external fields, and the matrix Riemann-Hilbert boundary value problem.
TL;DR: In this article, the infinity norm of the inverses of matrices of monotone type and totally positive matrices is estimated for both positive and non-positive matrices.
Abstract: We give estimates of the infinity norm of the inverses of matrices of monotone type and totally positive matrices.
TL;DR: In this article, the authors describe recent work that connects totally nonnegative matrices, quantum matrices and matrix Poisson varieties, and describe a survey article that connects these three objects of interest.
Abstract: In this survey article, we describe recent work that connects three separate objects of interest: totally nonnegative matrices; quantum matrices; and matrix Poisson varieties.
TL;DR: In this paper, the generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries, and the structure of the GSF is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix.
Abstract: In this paper, the concept of generalized spectral function is introduced for finite- order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.
TL;DR: In this article, it was shown that all eigenvalues of such matrices are of the form r ζ, where r is a nonnegative real number and ζ is a p th root of unity, where p is the period of the matrix, computed from the distance between the bands.
TL;DR: In this paper, Johnson and Sutton showed that for non-singular acyclic matrices, the maximum size of a P-set is at most n − 1 (or n) P-vertices.
Abstract: Let A be an n by n real symmetric matrix, and let α be a non-empty subset of {1, …, n}. The principal submatrix of A obtained by removing rows and columns indexed by α is denoted by A(α). If the nullity of A(α) is |α| more than that of A, then α is called a P-set of A, and the indices in α are called P-vertices of A. This article concerns P-vertices and P-sets of non-singular acyclic matrices A when the graph T of A is a path or a star. It was shown in [C.R. Johnson and B.D. Sutton, Hermitian matrices, eigenvalue multiplicities, and eigenvector components, SIAM J. Matrix Anal. Appl. 26 (2004), pp. 390–399] that each singular acyclic matrix of order n has at most n − 2 P-vertices. In this article we show that this does not hold for non-singular acyclic matrices by constructing non-singular acyclic matrices, whose graphs are T, having n − 1 (or n) P-vertices (these matrices also achieve the maximum size of a P-set over non-singular acyclic matrices whose graphs are T). In addition, it is shown that if n is ...
TL;DR: A formula is found for the entries of the inverse quantum Hilbert matrix of Hankel matrices corresponding to certain little q-Jacobi polynomials when |q|<1, and for the special value q=(1-5)(1+5) they are closely related to Hankel matrix of reciprocal Fibonacci numbers called Filbert matrices.
03 Aug 2009
TL;DR: This paper introduces matrix representations of algebraic curves and surfaces for Computer Aided Geometric Design and shows how to manipulate these representations by proposing a dedicated algorithm to address the curve/surface intersection problem by means of numerical linear algebra techniques.
Abstract: In this paper, we introduce matrix representations of algebraic curves and surfaces for Computer Aided Geometric Design (CAGD). The idea of using matrix representations in CAGD is quite old. The novelty of our contribution is to enable non square matrices, extension which is motivated by recent research in this topic. We show how to manipulate these representations by proposing a dedicated algorithm to address the curve/surface intersection problem by means of numerical linear algebra techniques.
21 Oct 2009
TL;DR: In this paper, the authors describe the geometry of systems of linear equations in R2 and R3 matrices and Echelon Form Gaussian Elimination and Reduced Row Echelan Form.
Abstract: Vectors Vectors in Rn The Inner Product and Norm Spanning Sets Linear Independence Bases Subspaces Summary Systems of Equations The Geometry of Systems of Equations in R2 and R3 Matrices and Echelon Form Gaussian Elimination Computational Considerations-Pivoting Gauss-Jordan Elimination and Reduced Row Echelon Form Ill-Conditioned Systems of Linear Equations Rank and Nullity of a Matrix Systems of m Linear Equations in n Unknowns Matrix Algebra Addition and Subtraction of Matrices Matrix-Vector Multiplication The Product of Two Matrices Partitioned Matrices Inverses of Matrices Elementary Matrices The LU Factorization Eigenvalues, Eigenvectors, and Diagonalization Determinants Determinants and Geometry The Manual Calculation of Determinants Eigenvalues and Eigenvectors Similar Matrices and Diagonalization Algebraic and Geometric Multiplicities of Eigenvalues The Diagonalization of Real Symmetric Matrices The Cayley-Hamilton Theorem (a First Look)/the Minimal Polynomial Vector Spaces Vector Spaces Subspaces Linear Independence and the Span Bases and Dimension Linear Transformations Linear Transformations The Range and Null Space of a Linear Transformation The Algebra of Linear Transformations Matrix Representation of a Linear Transformation Invertible Linear Transformations Isomorphisms Similarity Similarity Invariants of Operators Inner Product Spaces Complex Vector Spaces Inner Products Orthogonality and Orthonormal Bases The Gram-Schmidt Process Unitary Matrices and Orthogonal Matrices Schur Factorization and the Cayley-Hamilton Theorem The QR Factorization and Applications Orthogonal Complements Projections Hermitian Matrices and Quadratic Forms Linear Functionals and the Adjoint of an Operator Hermitian Matrices Normal Matrices Quadratic Forms Singular Value Decomposition The Polar Decomposition Appendix A: Basics of Set Theory Appendix B: Summation and Product Notation Appendix C: Mathematical Induction Appendix D: Complex Numbers Answers/Hints to Odd-Numbered Problems Index A Summary appears at the end of each chapter
TL;DR: P perturbation results for eigenvalues of a matrix pencil of Hankel matrices for which the elements are given by complex moments are presented, extended to the case that matrices have a block Hankel structure.
Abstract: In this paper, we present perturbation results for eigenvalues of a matrix pencil of Hankel matrices for which the elements are given by complex moments. These results are extended to the case that matrices have a block Hankel structure. The influence of quadrature error on eigenvalues that lie inside a given integral path can be reduced by using Hankel matrices of an appropriate size. These results are useful for discussing the numerical behavior of root finding methods and eigenvalue solvers which make use of contour integrals. Results from some numerical experiments are consistent with the theoretical results.
TL;DR: Based on the theory of first-order ordinary differential equations, a dual relation between two solutions in the dual local coordinates for a single layer in the laminate is derived, which is further arranged in a manner that can avoid the numerical instability usually encountered in the state space method as discussed by the authors.
TL;DR: Given a primitive stochastic matrix, an upper bound on the moduli of its non-Perron eigenvalues is provided in terms of the weights of the cycles in the directed graph associated with the matrix.
Abstract: Given a primitive stochastic matrix, we provide an upper bound on the moduli of its non-Perron eigenvalues. The bound is given in terms of the weights of the cycles in the directed graph associated with the matrix. The bound is attainable in general, and we characterize a special case of equality when the stochastic matrix has a positive row. Applications to Leslie matrices and to Google-type matrices are also considered.
Journal Article•
TL;DR: The weight of the factors which have influence on the result can be figured out by this method and the optimal schema of the orthogonal design and the order of the factor can be easily determined.
Abstract: This paper introduces a method of matrix analysis of orthogonal designThe weight of the factors which have influence on the result can be figured out by this methodAccording to the weight,the optimal schema of the orthogonal design and the order of the factor can be easily determined
TL;DR: In this article, an accurate equivalent circuit model to represent a 2D eddy-current magnetic field was presented, where the magnetic field can be coupled with stranded windings and solid conductors.
Abstract: We present an accurate equivalent circuit model to represent a 2-D eddy-current magnetic field. The magnetic field can be coupled with stranded windings and solid conductors. For use in developing formulations of impedance computation, we present two systematic matrix analysis methods based on the system equations, one using the loop method and the other using the nodal method. In the model, each solid conductor is represented by circuit branches. With this approach, the effect of eddy currents can be fully included. We also present the formulations for the computation of the total power loss for magnetic field-circuit coupled problems. We discuss a common mistake in many applications.
TL;DR: In this article, the Stenzel condition for the product of two skew-symmetric matrices was shown to be equivalent to the condition that every elementary divisor corresponding to a nonzero eigenvalue of C occurs an even number of times.
Abstract: We show that the product C of two skew-Hamiltonian matrices obeys the Stenzel conditions. If at least one of the factors is nonsingular, then the Stenzel conditions amount to the requirement that every elementary divisor corresponding to a nonzero eigenvalue of C occurs an even number of times. The same properties are valid for the product of two skew-pseudosymmetric matrices. We observe that the method proposed by Van Loan for computing the eigenvalues of real Hamiltonian and skew-Hamiltonian matrices can be extended to complex skew-Hamiltonian matrices. Finally, we show that the computation of the eigenvalues of a product of two skew-symmetric matrices reduces to the computation of the eigenvalues of a similar skew-Hamiltonian matrix. Bibliography: 8 titles.
28 Jul 2009
TL;DR: It is, however, non-trivial to retrieve the structural description of the matrix resulting from these operations, so an abstract matrix is defined as an encoding of support function combinations that enables simple recovery of the structural properties.
Abstract: Classes of matrices are often presented with symbolic dimensions using a mixture of terms and ellipsis symbols to describe their internal structure. While working with such classes of matrices is everyday mathematical practice, it has little automated support. We describe an algebraic encoding of such matrices in terms of support functions and define the corresponding addition and multiplication algorithms. It is, however, non-trivial to retrieve the structural description of the matrix resulting from these operations. We therefore define an abstract matrix as an encoding of support function combinations that enables simple recovery of the structural properties. This allows us to define arithmetic algorithms for abstract matrices as extensions of those for support function combinations using a normalising term rewrite system.