Showing papers on "Matrix analysis published in 2008"
Book•
01 Nov 2008
TL;DR: Finite Non-Negative Matrices as mentioned in this paper are a generalization of finite stochastic matrices, and finite non-negative matrices have been studied extensively in the literature.
Abstract: Finite Non-Negative Matrices.- Fundamental Concepts and Results in the Theory of Non-negative Matrices.- Some Secondary Theory with Emphasis on Irreducible Matrices, and Applications.- Inhomogeneous Products of Non-negative Matrices.- Markov Chains and Finite Stochastic Matrices.- Countable Non-Negative Matrices.- Countable Stochastic Matrices.- Countable Non-negative Matrices.- Truncations of Infinite Stochastic Matrices.
2,855 citations
TL;DR: Subspace sampling as discussed by the authors is a sampling method for low-rank matrix decompositions with relative error guarantees. But it is not known whether such a matrix decomposition exists in general.
Abstract: Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and/or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an $m\times n$ matrix $A$ and a rank parameter $k$. In our first algorithm, $C$ is chosen, and we let $A'=CC^+A$, where $C^+$ is the Moore-Penrose generalized inverse of $C$. In our second algorithm $C$, $U$, $R$ are chosen, and we let $A'=CUR$. ($C$ and $R$ are matrices that consist of actual columns and rows, respectively, of $A$, and $U$ is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least $1-\delta$, $\|A-A'\|_F\leq(1+\epsilon)\,\|A-A_k\|_F$, where $A_k$ is the “best” rank-$k$ approximation provided by truncating the SVD of $A$, and where $\|X\|_F$ is the Frobenius norm of the matrix $X$. The number of columns of $C$ and rows of $R$ is a low-degree polynomial in $k$, $1/\epsilon$, and $\log(1/\delta)$. Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple and they take time of the order needed to approximately compute the top $k$ singular vectors of $A$. The technical crux of our analysis is a novel, intuitive sampling method we introduce in this paper called “subspace sampling.” In subspace sampling, the sampling probabilities depend on the Euclidean norms of the rows of the top singular vectors. This allows us to obtain provable relative-error guarantees by deconvoluting “subspace” information and “size-of-$A$” information in the input matrix. This technique is likely to be useful for other matrix approximation and data analysis problems.
398 citations
TL;DR: In this article, a matrix perturbation approach was used to study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n and those of the limiting population PCA as n → oo.
Abstract: Principal component analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, and those of the limiting population PCA as n → oo. As in machine learning, we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample PCA and population PCA under a spiked covariance model. In addition, we also consider the relation between finite sample PCA and the asymptotic results in the joint limit p, n → ∞, with p/n = c. We present a matrix perturbation view of the "phase transition phenomenon," and a simple linear-algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit. Moreover, our analysis also applies for finite p, n where we show that although there is no sharp phase transition as in the infinite case, either as a function of noise level or as a function of sample size n, the eigenvector of sample PCA may exhibit a sharp "loss of tracking," suddenly losing its relation to the (true) eigenvector of the population PCA matrix. This occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise, whose eigenvector points in a random direction.
289 citations
Book•
26 Mar 2008
TL;DR: In this article, the authors give a thorough treatment of the theory of matrix functions and numerical methods for computing them and present a MATLAB toolbox providing implementations of the key algorithms.
Abstract: The only book devoted exclusively to matrix functions, this research monograph gives a thorough treatment of the theory of matrix functions and numerical methods for computing them. The author s elegant presentation focuses on the equivalent definitions of f(A) via the Jordan canonical form, polynomial interpolation, and the Cauchy integral formula, and features an emphasis on results of practical interest and an extensive collection of problems and solutions. Functions of Matrices: Theory and Computation is more than just a monograph on matrix functions; its wide-ranging content including an overview of applications, historical references, and miscellaneous results, tricks, and techniques with an f(A) connection makes it useful as a general reference in numerical linear algebra. Other key features of the book include development of the theory of conditioning and properties of the Frchet derivative; an emphasis on the Schur decomposition, the block Parlett recurrence, and judicious use of Pad approximants; the inclusion of new, unpublished research results and improved algorithms; a chapter devoted to the f(A)b problem; and a MATLAB toolbox providing implementations of the key algorithms. Audience: This book is for specialists in numerical analysis and applied linear algebra as well as anyone wishing to learn about the theory of matrix functions and state of the art methods for computing them. It can be used for a graduate-level course on functions of matrices and is a suitable reference for an advanced course on applied or numerical linear algebra. It is also particularly well suited for self-study. Contents: List of Figures; List of Tables; Preface; Chapter 1: Theory of Matrix Functions; Chapter 2: Applications; Chapter 3: Conditioning; Chapter 4: Techniques for General Functions; Chapter 5: Matrix Sign Function; Chapter 6: Matrix Square Root; Chapter 7: Matrix pth Root; Chapter 8: The Polar Decomposition; Chapter 9: Schur-Parlett Algorithm; Chapter 10: Matrix Exponential; Chapter 11: Matrix Logarithm; Chapter 12: Matrix Cosine and Sine; Chapter 13: Function of Matrix Times Vector: f(A)b; Chapter 14: Miscellany; Appendix A: Notation; Appendix B: Background: Definitions and Useful Facts; Appendix C: Operation Counts; Appendix D: Matrix Function Toolbox; Appendix E: Solutions to Problems; Bibliography; Index.
217 citations
TL;DR: The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method and the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily.
Abstract: The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally treelike, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, showing excellent agreement.
147 citations
TL;DR: In this paper, the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles is given in the form of a matrix product representation.
Abstract: We find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles. The solution is in the form of a matrix product representation where the matrices for a system of N species are defined recursively in terms of the matrices for a system of N-1 species. A complete proof is given, based on the quadratic relations verified by these matrices. This matrix product construction is interpreted in terms of the action of a transfer matrix.
109 citations
TL;DR: In this paper, the authors rigorously study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals, which include the Sierpinski gasket and other 3n-gaskets.
Abstract: We rigorously study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals, which include the Sierpinski gasket and other 3n-gaskets We consider the classical Laplacian on fractals which generalizes the usual one-dimensional second derivative, is the generator of the self-similar diffusion process, and has possible applications as the quantum Hamiltonian We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly We support our theoretical analysis by symbolic and numerical computations Our analysis, in particular, allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues (ie, integrated density of states) We consider such examples as the level-3 Sierpinski gasket, a fractal 3-tree, and the diamond fractal
109 citations
Book•
06 Aug 2008TL;DR: In this paper, the Konig Digraph of a Matrix Partitioned Matrices Powers of Matrices Matrix Powers and Digraphs Circulant Matrices Permutations with Restrictions Determinants Definition of the Determinant properties of determinants A Special Determinante Formula Classical Definition of DeterminANT Laplace Development of the determinant Matrix Inverses Adjoint and Its Determinent Inverse of a Square Matrix Graph-Theoretic Interpretation Systems of Linear Equations Solutions of Linear Systems Cramer's Formula Solving Linear Systems by Digraph
Abstract: Introduction Graphs Digraphs Some Classical Combinatorics Fields Vector Spaces Basic Matrix Operations Basic Concepts The Konig Digraph of a Matrix Partitioned Matrices Powers of Matrices Matrix Powers and Digraphs Circulant Matrices Permutations with Restrictions Determinants Definition of the Determinant Properties of Determinants A Special Determinant Formula Classical Definition of the Determinant Laplace Development of the Determinant Matrix Inverses Adjoint and Its Determinant Inverse of a Square Matrix Graph-Theoretic Interpretation Systems of Linear Equations Solutions of Linear Systems Cramer's Formula Solving Linear Systems by Digraphs Signal Flow Digraphs of Linear Systems Sparse Matrices Spectrum of a Matrix Eigenvectors and Eigenvalues The Cayley-Hamilton Theorem Similar Matrices and the JCF Spectrum of Circulants Nonnegative Matrices Irreducible and Reducible Matrices Primitive and Imprimitive Matrices The Perron-Frobenius Theorem Graph Spectra Additional Topics Tensor and Hadamard Product Eigenvalue Inclusion Regions Permanent and Sign-Nonsingular Matrices Applications Electrical Engineering: Flow Graphs Physics: Vibration of a Membrane Chemistry: Unsaturated Hydrocarbons Exercises appear at the end of each chapter.
107 citations
TL;DR: algebraic random matrices as discussed by the authors is a class of matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation.
Abstract: We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marcenko–Pastur law are special cases.
Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form.
In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.
95 citations
TL;DR: In this article, an extension of the diagrammatic rules in random matrix theory and applying it to nonhermitean random matrix models using the 1/N approximation was introduced, and a number of one-and two-point functions were evaluated on their holomorphic and nonholomorphic supports to leading order in 1 /N.
Abstract: We introduce an extension of the diagrammatic rules in random matrix theory and apply it to nonhermitean random matrix models using the 1/N approximation. A number of one- and two-point functions are evaluated on their holomorphic and nonholomorphic supports to leading order in 1/N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic correlations. Generic form for the two-point functions are obtained, generalizing the concept of macroscopic universality to nonhermitean random matrices. We show that the holomorphic and nonholomorphic one- and two-point functions condition the behavior of pertinent partition functions to order O(1/N). We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.
87 citations
TL;DR: In this article, a matrix is said to have the Perron-Frobenius property if its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector.
Abstract: A matrix is said to have the Perron-Frobenius propertyif its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such gener- alized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are proved. Similaritytransformations leaving such sets invariant are completelydescribed, and it is shown that a nonnilpotent matrix eventuallycapturing the Perron-Frobenius propertyis in fact a matrix that alreadyhas it.
TL;DR: In this paper, the authors proposed a modal analysis of complex three-dimensional framed structures, optionally with lumped masses and/or elastic supports, and established orthogonal conditions for different natural modes, and hence transient response analysis based on mode superposition.
TL;DR: This paper presents some new skew-Hadamard matrices of order 52 and improves the known lower bound on the number of the skew- hadamardMatrices of this order.
TL;DR: Necessary and sufficient conditions for positive realness of general transfer function matrices are derived in terms of eigenvalues of matrix functions of the state matrices representation of the LTI system.
Abstract: Necessary and sufficient conditions for positive realness of general transfer function matrices are derived. The conditions are expressed in terms of eigenvalues of matrix functions of the state matrices representation of the LTI system. Illustrative numerical examples are provided.
TL;DR: In this article, the authors explore an underlying geometric framework in which matrix multiplication naturally arises from the product of numbers in a geometric (Clifford) algebra, and all invariants of a linear operator become geometric invariants for the multivectors that they represent.
TL;DR: In this paper, the perturbation theory for purely imaginary eigenvalues of Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations is discussed and it is shown that there is a substantial difference in the behavior under these perturbants.
Abstract: The perturbation theory for purely imaginary eigenvaluesof Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations is discussed It is shown that there is a substantial difference in the behavior under these perturbations The perturbation of real eigenvalues of real skew-Hamiltonian matrices under structured perturbations is discussed as well and these results are used to analyze the properties of the URV method for computing the eigenvalues of Hamiltonian matrices
TL;DR: For the Chain of Matrices model, this article showed that the topological expansion of the free energy is, like for the 1 and 2-matrix model, given by the symplectic invariants of the associated spectral curve.
Abstract: We solve the loop equations to all orders in $1/N^2$, for the Chain of Matrices matrix model (with possibly an external field coupled to the last matrix of the chain). We show that the topological expansion of the free energy, is, like for the 1 and 2-matrix model, given by the symplectic invariants of the associated spectral curve. As a consequence, we find the double scaling limit explicitly, and we discuss modular properties, large $N$ asymptotics. We also briefly discuss the limit of an infinite chain of matrices (matrix quantum mechanics).
TL;DR: In this paper, the Shapley value and the associated game are represented algebraically by their coalitional matrices, called Shapley standard matrices and associated transformation matrix, respectively.
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TL;DR: In this paper, relative oscillation theory for Jacobi matrices is developed, which replaces nodes of solutions associated with one matrix by weighted nodes of Wronskians of solutions of two different matrices.
Abstract: We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by replacing nodes of solutions associated with one matrix by weighted nodes of Wronskians of solutions of two different matrices.
TL;DR: In this paper, a comparison of two approaches to analyze the periodic arrays of left-handed metamaterials by field simulations in a single unit cell is presented, and the authors show how these results can be transformed into each other, and present a quantitative analysis of the impact of higher order modes on the effective array properties.
Abstract: We propose a comparison of two approaches to analyze the periodic arrays of left-handed metamaterials by field simulations in a single unit cell. First, a scattering matrix analysis in time domain is able to produce broadband results. The set of transfer functions between a small number of input and output ports of the unit cell is able to describe its macroscopic behavior, where, however, the appearance of higher order modes is systematically neglected. As a reference result with respect to this issue, an eigenvalue analysis with periodic boundary conditions provides one point of the dispersion relation of the array per simulation run. We show how these results can be transformed into each other, and present a quantitative analysis of the impact of higher order modes on the effective array properties.
01 Jan 2008
TL;DR: This work focuses on dense matrices of the form AD 2 A T, where A is an m◊n matrix (m n) and D is an n ◊ n diagonal matrix, and develops in this work ecient algorithms for linear and nonlinear programming problems.
Abstract: Matrix computations are expensive, and GPUs have the potential to deliver results at reduced cost by exploiting parallel computation. We focus on dense matrices of the form AD 2 A T , where A is an m◊n matrix (m n) and D is an n ◊ n diagonal matrix. Many important numerical problems require solving linear systems of equations involving matrices of this form. These problems include normal equations approaches to solving linear least squares and weighted linear least squares problems, and interior point algorithms for linear and nonlinear programming problems. We develop in this work ecient
TL;DR: A survey of the properties of conjugate normal matrices is given in this paper, where a list of more than forty conditions are given, each of which is equivalent to A being a normal.
TL;DR: In this article, single real transformation matrices of typical symmetrical systems with double three-phase and two parallel double-three-phase transmission lines were tested using linear combinations of the Clarke's matrix elements.
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TL;DR: A special decomposition of numeric genetic matrices reveals their close relations with a family of 8-dimensional hypercomplex numbers (not Cayley's octonions). Some hypothesis and thoughts are formulated on the basis of these phenomenological facts.
Abstract: The matrix form of the presentation of the genetic code is described as the cognitive form to analyze structures of the genetic code A similar matrix form is utilized in the theory of signal processing The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet This matrix in the third Kronecker power is the (8*8)-matrix, which contains 64 triplets Peculiarities of the degeneracy of the vertebrate mitochondria genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing, spectral analysis, quantum mechanics and quantum computers A special decomposition of numeric genetic matrices reveals their close relations with a family of 8-dimensional hypercomplex numbers (not Cayley's octonions) Some hypothesis and thoughts are formulated on the basis of these phenomenological facts
TL;DR: In this article, the absolute values of entries of matrix valued functions of infinite matrices, as well as two-sided estimates for the entries of matrices of infinite M-matrices (monotone matrices) are derived.
Abstract: Sharp upper estimates for the absolute values of entries of matrix valued functions of infinite matrices, as well as two sided estimates for the entries of matrix valued functions of infinite M-matrices (monotone matrices) are derived. They give us bounds for the lattice norms of matrix valued functions and positivity conditions for functions of M-matrices. In addition, some results on perturbations and comparison of matrix functions are proved. Applications of the obtained estimates to the Hille-Tamarkin matrices and differential equations are also discussed.
TL;DR: The main idea underlying the presented techniques is to exploit the special block structure of a Hamiltonian and skew-symmetric matrix to gain a cheaper computation of the functions.
TL;DR: In this article, the density of complex eigenvalues for random matrices of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are distributed on the group $\mathrm{U(N)}$ according to the Haar measure is derived.
Abstract: Given any fixed $N \times N$ positive semi-definite diagonal matrix $G\ge 0$ we derive the explicit formula for the density of complex eigenvalues for random matrices $A$ of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are distributed on the group $\mathrm{U(N)}$ according to the Haar measure.
TL;DR: An algebraic theory of rigidity for finite-element matrices is presented and it is shown that one particular construction algorithm for fretsaw extensions generates matrices that can be factored with essentially no fill.
Abstract: We present an algebraic theory of rigidity for finite-element matrices The theory provides a formal algebraic definition of finite-element matrices; notions of rigidity of finite-element matrices and of mutual rigidity between two such matrices; and sufficient conditions for rigidity and mutual rigidity We also present a novel sparsification technique, called fretsaw extension, for finite-element matrices We show that this sparsification technique generates matrices that are mutually rigid with the original matrix We also show that one particular construction algorithm for fretsaw extensions generates matrices that can be factored with essentially no fill This algorithm can be used to construct preconditioners for finite-element matrices Both our theory and our algorithms are applicable to a wide range of finite-element matrices, including matrices arising from finite-element discretizations of both scalar and vector partial differential equations (eg, electrostatics and linear elasticity) Both the theory and the algorithms are purely algebraic-combinatorial They manipulate only the element matrices and are oblivious to the geometry, the material properties, and the discretization details of the underlying continuous problem
TL;DR: In this article, a short proof of Brody's conjecture is presented, drawing upon some theorems obtained in recent years in the literature on empirical spectral distribution of random matrices, and discusses briefly some related issues.
Abstract: Brody (1997) notices that for large random Leontief matrices, namely non-negative square matrices with all entries i.i.d., the ratio between the subdominant eigenvalue (in modulus) and the dominant eigenvalue declines generically to zero at a speed of the square root of the size of the matrix as the matrix size goes to infinity. Since then, several studies have been published in this journal in attempting to rigorously verify Brody's conjecture. This short article, drawing upon some theorems obtained in recent years in the literature on empirical spectral distribution of random matrices, offers a short proof of Brody's conjecture, and discusses briefly some related issues.
TL;DR: In this article, the eigenvalues of many-body systems interacting by two-body forces as well as those of random matrices were investigated and a strong linear correlation between eigen values and diagonal matrix elements if both of them are sorted from the smaller values to larger ones.
Abstract: We investigate eigenvalues of many-body systems interacting by two-body forces as well as those of random matrices. For two-body random ensemble, we find a strong linear correlation between eigenvalues and diagonal matrix elements if both of them are sorted from the smaller values to larger ones. By using this linear correlation we are able to predict reasonably all eigenvalues of a given Hamiltonian matrix without complicated iterations. For Gaussian orthogonal ensemble matrices, the hyperbolic tangent function improves the accuracy of predicted eigenvalues near the minimum and maximum.