Showing papers on "Matrix analysis published in 2011"

Matrix Algebra From a Statistician's Perspective

Abstract: Preface. - Matrices. - Submatrices and partitioned matricies. - Linear dependence and independence. - Linear spaces: row and column spaces. - Trace of a (square) matrix. - Geometrical considerations. - Linear systems: consistency and compatability. - Inverse matrices. - Generalized inverses. - Indepotent matrices. - Linear systems: solutions. - Projections and projection matrices. - Determinants. - Linear, bilinear, and quadratic forms. - Matrix differentiation. - Kronecker products and the vec and vech operators. - Intersections and sums of subspaces. - Sums (and differences) of matrices. - Minimzation of a second-degree polynomial (in n variables) subject to linear constraints. - The Moore-Penrose inverse. - Eigenvalues and Eigenvectors. - Linear transformations. - References. - Index.

Fluctuations of eigenvalues of random normal matrices

Abstract: In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane—the “droplet.” We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.

Fluctuations of the Extreme Eigenvalues of Finite Rank Deformations of Random Matrices

Abstract: Consider a deterministic self-adjoint matrix $X_n$ with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalised eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix $X_n$ so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when $X_n$ is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.

Combinatorial and Graph-Theoretical Problems in Linear Algebra

Abstract: This volume aims to gather information from both those who work on linear algebra problems in which combinatorial or graph-theoretical analysis is a major component, and those that work on combinatorial or graph-theoretical problems for which linear algebra is a major tool. Specific topics covered in the papers include matrix problems and results in symbolic dynamics, block-triangular decompositions of mixed matrices, algebraic and geometric properties of Laplacian matrices of graphs, the use of eigenvalues in combinatorial optimization, perturbation effects on rank and eigenvalues, and polynomial spaces.

A Unique “Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices

Abstract: This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear system only if the measurement matrix has a row-span intersecting the positive orthant. Focusing on two types of binary measurement matrices, Bernoulli 0-1 matrices and adjacency matrices of general expander graphs, we show that, in both cases, the support size of a unique nonnegative solution can grow linearly, namely O(n), with the problem dimension n . We also provide closed-form characterizations of the ratio of this support size to the signal dimension. For the matrix case, we show that under a necessary and sufficient condition for the linear compressed observations operator, there will be a unique positive semidefinite matrix solution to the compressed linear observations. We further show that a randomly generated Gaussian linear compressed observations operator will satisfy this condition with overwhelmingly high probability.

A Simplified Approach to Recovery Conditions for Low Rank Matrices

Abstract: Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including $\ell_1$ and nuclear norm minimization as well as $\ell_p$ minimization with $p<1$. These algorithms are known to succeed if certain conditions on the measurement map are satisfied. Proofs of robust recovery for matrices have so far been much more involved than in the vector case. In this paper, we show how several robust classes of recovery conditions can be extended from vectors to matrices in a simple and transparent way, leading to the best known restricted isometry and nullspace conditions for matrix recovery. Our results rely on the ability to "vectorize" matrices through the use of a key singular value inequality.

User-Friendly Tail Bounds for Matrix Martingales

Abstract: This report presents probability inequalities for sums of adapted sequences of random, self-adjoint matrices. The results frame simple, easily verifiable hypotheses on the summands, and they yield strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. The methods also specialize to sums of independent random matrices.

A method for generating realistic correlation matrices

Abstract: Simulating sample correlation matrices is important in many areas of statistics. Approaches such as generating Gaussian data and finding their sample correlation matrix or generating random uniform $[-1,1]$ deviates as pairwise correlations both have drawbacks. We develop an algorithm for adding noise, in a highly controlled manner, to general correlation matrices. In many instances, our method yields results which are superior to those obtained by simply simulating Gaussian data. Moreover, we demonstrate how our general algorithm can be tailored to a number of different correlation models. Using our results with a few different applications, we show that simulating correlation matrices can help assess statistical methodology.

Projection matrices, generalized inverse matrices, and singular value decomposition

Abstract: Fundamentals of Linear Algebra.- Projection Matrices.- Generalized Inverse Matrices.- Explicit Representations.- Singular Value Decomposition (SVD).- Various Applications.

On the eigenvalues of quaternion matrices

Abstract: This article is a continuation of the article [F. Zhang, Gersgorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2 × 2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Gersgorin.

Algorithm for Error-Controlled Simultaneous Block-Diagonalization of Matrices

Abstract: An algorithm is given for the problem of finding the finest simultaneous block-diagonalization of a given set of square matrices. This problem has been studied independently in the area of semidefinite programming and independent component analysis. The proposed algorithm considers the commutant algebra of the matrix *-algebra generated by the given matrices. It is simpler than other existing methods, and has the capability of controlling numerical errors. Some numerical examples are presented to demonstrate its merits.

The Restricted Isometry Property for block diagonal matrices

Abstract: In compressive sensing (CS), the Restricted Isometry Property (RIP) is a powerful condition on measurement operators which ensures robust recovery of sparse vectors is possible from noisy, undersampled measurements via computationally tractable algorithms. Early papers in CS showed that Gaussian random matrices satisfy the RIP with high probability, but such matrices are usually undesirable in practical applications due to storage limitations, computational considerations, or the mismatch of such matrices with certain measurement architectures. To alleviate some or all of these difficulties, recent research efforts have focused on structured random matrices. In this paper, we study block diagonal measurement matrices where each block on the main diagonal is itself a Gaussian random matrix. The main result of this paper shows that such matrices can indeed satisfy the RIP but that the requisite number of measurements depends on the coherence of the basis in which the signals are sparse. In the best case—for signals that are sparse in the frequency domain—these matrices perform nearly as well as dense Gaussian random matrices despite having many fewer nonzero entries.

On a Family of Circulant Matrices for Quasi-Cyclic Low-Density Generator Matrix Codes

Abstract: We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.

Calibration of Nodal Demand in Water Distribution Systems

Abstract: This paper presents an optimization methodology to calibrate the nodal demand in the model of water supply distribution systems. The matrix analysis has been carried out to get the sensitive coefficients matrix for the model of water distribution systems. Singular value decomposition is applied to calculate search director of optimization by solving the sensitive coefficients matrix, which is more efficient than the finite-difference method. Two cases are used to show the performance of this algorithm. The first case is a hypothetical water distribution network showing the calculation of the optimization vector. The second case is the application of real-life water distribution system in Hangzhou, China, which shows the determination of the cutoff singular value and the search step size in one-dimensional search. The calibrated model has been also validated using the measured data. The results have shown that the proposed algorithm is reliable and effective in real-time system.

Generalized parity-check matrices for SEC-DED codes with fixed parity

Abstract: Hsiao and extended Hamming parity-check matrices can be used to define systematic linear block codes for Single Error Correction-Double Error Detection (SEC-DED). Their fixed code word parity enables the construction of low density parity-check matrices and fast hardware implementations. Fixed code word parity is enabled by an all-one row in extended Hamming parity-check matrices or by the constraint that the modulo-2 sum of all rows is equal to the all-zero vector in Hsiao parity-check matrices. In this paper, we show that these two constraints are particular instantiations of a more general constraint which involves an arbitrary number of rows in the parity-check matrix. As a consequence, sparser parity-check matrices with faster hardware implementations can be found. Moreover, special instantiations of these matrices enable the detection of all triple-bit and quadruple-bit burst errors.

The influence of a matrix condition number on iterative methods' convergence

Abstract: We investigate condition numbers of matrices that appear during solving systems of linear equations. We consider iterative methods to solve the equations, namely Jacobi and Gauss-Seidel methods. We examine the influence of the condition number on convergence of these iterative methods. We study numerical aspects of relations between the condition number and the size of the matrix and the number of iterations experimentally. We analyze random matrices, the Hilbert matrix and a strictly diagonally dominant matrix.

Real second-order freeness and the asymptotic real second-order freeness of several real matrix ensembles

Abstract: We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, three real models of random matrices, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. We use a combinatorial approach to the matrix calculations similar to the genus expansion for complex random matrices, but in which nonorientable surfaces appear, demonstrating the commonality between the real models and the distinction from their complex analogues, motivating this distinct definition. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears.

A Matrix Approach for General Higher Order Linear Recurrences

Abstract: We consider k sequences of generalized order-k linear recurrences with arbitrary initial conditions and coecients, and we give their generalized Binet formulas and generating functions. We also obtain a new matrix method to derive explicit formulas for the sums of terms of the k sequences. Further, some relationships between determinants of certain Hessenberg matrices and the terms of these sequences are obtained.

Efficient computation with structured matrices and arithmetic expressions

Abstract: Designing efficient code in practice for a given computation is a hard task. In this thesis, we tackle this issue in two different situations. The first part of the thesis introduces some algorithmic improvements in structured linear algebra. We first show how to extend an algorithm by Cardinal for inverting Cauchy-like matrices to the other common structures. This approach, which mainly relies on products of the type "structured matrix × matrix", leads to a theoretical speed-up of a factor up to 7 that we also observe in practice. Then, we extend some works on Toeplitz-like matrices and prove that, for any of the common structures, the product of an n×n structured matrix of displacement rank α by an n×α matrix can be computed in O(α^(ω-1)n). This leads to direct inversion algorithms in O(α^(ω-1)n) , that do not rely on a reduction to the Toeplitz-like case. The second part of the thesis deals with the implementation of arithmetic expressions. This topic raises several issues like finding the minimum number of operations, and maximizing the speed or the accuracy when using some finite-precision arithmetic. Making use of the inductive nature of arithmetic expressions enables the design of algorithms that help to answer such questions. We thus present a set of algorithms for generating evaluation schemes, counting them, and optimizing them according to one or several criteria. These algorithms are part of a library that we have developed and used, among other things, in order to decrease the running time of a code generator for a mathematical library, and to study optimality issues about the evaluation of a small degree polynomial with scalar coefficients at a matrix point.

Non-Gaussian Limiting Laws for the Entries of Regular Functions of the Wigner Matrices

Abstract: This paper is a continuation of our paper "Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices", J. Stat. Phys. (134), 147--159 (2009), in which we proved the Central Limit Theorem for the matrix elements of differential functions of the real symmetric random Gaussian matrices (GOE). Here we consider the real symmetric random Wigner matrices having independent (modulo symmetry conditions) but not necessarily Gaussian entries. We show that in this case the matrix elements of sufficiently smooth functions of these random matrices have in general another limiting law which coincides essentially with the probability law of matrix entries.

Numerical-symbolic exact irreducible decomposition of cyclic-12

Abstract: In 1992, Goran Bjorck and Ralf Froberg completely characterized the solution set of cyclic-8. In 2001, Jean-Charles Faugere determined the solution set of cyclic-9, by computer algebra methods and Grobner basis computation. In this paper, a new theory in matrix analysis of rank-deficient matrices together with algorithms in numerical algebraic geometry enables us to present a symbolic-numerical algorithm to derive exactly the defining polynomials of all prime ideals of positive dimension in primary decomposition of cyclic-12. Empirical evidence together with rigorous proof establishes the fact that the positive-dimensional solution variety of cyclic-12 just consists of 72 quadrics of dimension one.

Eigenvalue Localization Refinements for Matrices Related to Positivity

Abstract: Eigenvalue localization results and methods for matrices with constant row or column sum are provided, together with the numerical examples that show the efficiency of the proposed methods. The extension of the results to other classes of matrices is additionally analyzed.