Showing papers on "Matrix analysis published in 2017"
Book•
05 Oct 2017TL;DR: In this article, the authors present the theory, methods and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis.
Abstract: This balanced and comprehensive study presents the theory, methods and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis in a completely new light Alongside the core subjects in matrix analysis, such as singular value analysis, the solution of matrix equations and eigenanalysis, the author introduces new applications and perspectives that are unique to this book The very topical subjects of gradient analysis and optimization play a central role here Also included are subspace analysis, projection analysis and tensor analysis, subjects which are often neglected in other books Having provided a solid foundation to the subject, the author goes on to place particular emphasis on the many applications matrix analysis has in science and engineering, making this book suitable for scientists, engineers and graduate students alike
613 citations
TL;DR: In this article, the authors developed a method for deriving local laws for a large class of random matrices, which can be applied to many matrix models built from sums and products of deterministic or independent random matrix matrices.
Abstract: We develop a new method for deriving local laws for a large class of random matrices It is applicable to many matrix models built from sums and products of deterministic or independent random matrices In particular, it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity For definiteness, we present the method for sample covariance matrices of the form , where T is deterministic and X is random with independent entries We prove that with high probability the resolvent of Q is close to a deterministic matrix, with an optimal error bound and down to optimal spectral scales As an application, we prove the edge universality of Q by establishing the Tracy–Widom–Airy statistics of the eigenvalues of Q near the soft edges This result applies in the single-cut and multi-cut cases Further applications include the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations; they will appear elsewhere We also apply our method to Wigner matrices whose entries have arbitrary expectation, ie we consider $$W+A$$
where W is a Wigner matrix and A a Hermitian deterministic matrix We prove the anisotropic local law for $$W+A$$
and use it to establish edge universality
123 citations
TL;DR: In this article, a data-driven method is proposed to reveal the correlations between the factors and the system status through statistical properties of data, which is based on the random matrix theory.
Abstract: The operating status of power systems is influenced by growing varieties of factors, resulting from the developing sizes and complexity of power systems. In this situation, the model-based methods need to be revisited. A data-driven method, as the novel alternative on the other hand, is proposed in this paper. It reveals the correlations between the factors and the system status through statistical properties of data. An augmented matrix as the data source is the key trick for this method and is formulated by two parts: 1) status data as the basic part; and 2) factor data as the augmented part. The random matrix theory is applied as the mathematical framework. The linear eigenvalue statistics, such as the mean spectral radius, are defined to study data correlations through large random matrices. Compared with model-based methods, the proposed method is inspired by a pure statistical approach without a prior knowledge of operation and interaction mechanism models for power systems and factors. In general, this method is direct in analysis, robust against bad data, universal to various factors, and applicable for real-time analysis. A case study based on the standard IEEE 118-bus system validates the proposed method.
94 citations
TL;DR: A review of transfer matrix and matrix product operators can be found in this article, where the authors present a mixture of classic results, presented from the point of view of tensor networks, and new results.
Abstract: Transfer matrices and matrix product operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic overview of applications, exact results, and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.
60 citations
TL;DR: This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.
Abstract: Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.
33 citations
TL;DR: In this article, the comparative index was used to obtain new Sturmian separation theorems as well as new and optimal estimates for left and right proper focal points of conjoined bases of linear Hamiltonian systems on bounded intervals.
29 citations
TL;DR: In this paper, an approach based on polynomial matrix theory for formulating the equations of motion and determining the response of multi-degree-of-freedom (MDOF) linear dynamical systems with singular matrices and subject to linear constraints is developed.
27 citations
TL;DR: Computational procedures are developed which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations as a consequence of some new relationships between the problem of solving matrix equations and theproblem of numerical computation of generalizedInverses.
Abstract: Conditions for the existence and representations of -, -, and -inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.
23 citations
05 Mar 2017
TL;DR: This study introduces an a priori statistical model for time-series amplitudes of source spectrograms, employing a new frequency-wise sparse regularization using estimates from the Bayesian postfilter to enhance the modeling accuracy.
Abstract: In this paper, we propose a new blind source separation (BSS) method based on independent low-rank matrix analysis (ILRMA) with novel sparse regularization. ILRMA is a recently proposed BSS algorithm that simultaneously estimates a demixing matrix and source spectrogram models based on nonnegative matrix factorization (NMF). To improve the separation accuracy and stability, an additional constraint such as sparseness is needed but there have been no studies on this so far. In this study, we introduce an a priori statistical model for time-series amplitudes of source spectrograms, employing a new frequency-wise sparse regularization using estimates from the Bayesian postfilter to enhance the modeling accuracy. This regularization results in a bilevel optimization problem that consists of the estimation of a sparsity-emphasized source model using NMF and the separation of sources by ILRMA. In this paper, we present two approximated optimization schemes and their combination for performing regularized ILRMA. The efficacy of the proposed method is confirmed in a BSS experiment.
23 citations
TL;DR: It is shown that the linear layers with a well-chosen nonlinear layer can provide sufficient security against differential and linear cryptanalysis and lower bounds on the number of active S-boxes are studied.
Abstract: Near-MDS matrices provide better trade-offs between security and efficiency compared to constructions based on MDS matrices, which are favored for hardwareoriented designs. We present new designs of lightweight linear diffusion layers by constructing lightweight near-MDS matrices. Firstly generic n × n near-MDS circulant matrices are found for 5 ≤ n ≤9. Secondly, the implementation cost of instantiations of the generic near-MDS matrices is examined. Surprisingly, for n = 7 , 8, it turns out that some proposed near-MDS circulant matrices of order n have the lowest XOR count among all near-MDS matrices of the same order. Further, for n = 5 , 6, we present near-MDS matrices of order n having the lowest XOR count as well. The proposed matrices, together with previous construction of order less than five, lead to solutions of n × n near-MDS matrices with the lowest XOR count over finite fields F 2 m for 2 ≤ n ≤ 8 and 4 ≤ m ≤ 2048. Moreover, we present some involutory near-MDS matrices of order 8 constructed from Hadamard matrices. Lastly, the security of the proposed linear layers is studied by calculating lower bounds on the number of active S-boxes. It is shown that our linear layers with a well-chosen nonlinear layer can provide sufficient security against differential and linear cryptanalysis.
20 citations
TL;DR: Bahsi and Solak as discussed by the authors considered k-circulant matrices with arithmetic sequence and investigated the eigenvalues, determinants and Euclidean norms of such matrices.
Abstract: Let k be a nonzero complex number. In this paper we consider k - circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k=1 , the inverses of such ( invertible ) matrices are obtained (in a way different from the way presented in the paper: M. Bahsi and S. Solak , On the Circulant Matrices with Arithmetic Sequence , Int. J. Contemp . Math. Sci. 5 (25) (2010), 1213-1222, and the Moore-Penrose inverses of such (singular) matrices are derived.
TL;DR: The limiting distribution of the eigen values of a general Fisher matrix is found and a central limit theorem is established for a wide class of functionals of these eigenvalues.
Abstract: Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two covariance matrices, or testing the independence between sub-groups of a multivariate random vector. Most of the existing work on random Fisher matrices deals with a particular situation where the population covariance matrices are equal. In this paper, we consider general Fisher matrices with arbitrary population covariance matrices and develop their spectral properties when the dimensions are proportionally large compared to the sample size. The paper has two main contributions: first the limiting distribution of the eigenvalues of a general Fisher matrix is found and second, a central limit theorem is established for a wide class of functionals of these eigenvalues. Applications of the main results are also developed for testing hypotheses on high-dimensional covariance matrices.
01 Jun 2017
TL;DR: In this paper, the authors established some analytical formulas for calculating ranks and inertias of linear unbiased estimators' covariance matrices under a general linear model, and using these formulas in the comparison of covariance matrix of BLUEs with other types of estimators.
Abstract: Best linear unbiased estimators (BLUEs) of unknown parameters under linear models have minimum covariance matrices in the Lowner partial ordering among all linear unbiased estimators of the unknown parameters. Hence, BLUEs’ covariance matrices are usually used as a criterion to compare optimality with other types of estimator. During this work, people often need to establish certain equalities and inequalities for BLUEs’ covariance matrices, and use them in statistical inference of regression models. This paper aims at establishing some analytical formulas for calculating ranks and inertias of BLUEs’ covariance matrices under general linear model, and using these formulas in the comparison of covariance matrices of BLUEs with other types of estimator. This is in fact a mathematical work, and some new tools in matrix analysis are essentially utilized.
TL;DR: The method of random skewers (RS method) has been used to compare a matrix to an a priori vector or to compare two matrices to find the average angle of the response vectors corresponding from another matrix.
Abstract: Ecological and evolutionary studies are often concerned with the properties of covariance matrices. The method of random skewers (RS method) has been used compare a matrix to an a priori vector or to compare two matrices. The method involves multiplying a matrix by many random vectors drawn from a uniform distribution over all possible vector directions. The comparisons are usually made using the average angle (or cosine) of the response vectors to an a priori vector or to the response vectors corresponding from another matrix. Angles are usually constrained to the interval 0°–90° because the distribution of response vectors is bipolar bimodal. The size of the average angle or cosine depends strongly on the relative sizes of the eigenvalues (especially the first). The distribution of angles between pairs of response vectors from two covariance matrices is more complicated because it depends on the differences in orientation of the eigenvectors and the relative sizes of the eigenvalues of the both matrices. The average absolute value of the angles between these pairs of response vectors depends on the relative sizes of the eigenvalues of the matrices making it difficult to interpret its meaning without knowledge of the eigenvalues and eigenvectors of the two matrices. Thus, it is simpler to just directly compare matrices in terms of these quantities.
TL;DR: In this article, the inverse of conjugate-Toeplitz (CT) matrices was shown to be the sum of products of lower and upper triangular matrices, and the decomposition of the inverse was obtained.
Abstract: In this paper, we consider the conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices. It is proved that the inverse of these special matrices can be expressed as the sum of products of lower and upper triangular matrices. Firstly, we get access to the explicit inverse of conjugate-Toeplitz matrix. Secondly, the decomposition of the inverse is obtained. Similarly, the formulae and the decomposition on inverse of conjugate-Hankel are provided. Thirdly, the stability of the inverse formulae of CT and CH matrices are discussed. Finally, examples are provided to verify the feasibility of the algorithms provided in this paper.
TL;DR: In this article, it was shown that for every trace-zero matrix A over an arbitrary field, there is a non-negative integer p such that the extended matrix is the sum of three square-zero matrices (more precisely, one can simply take p as the number of rows of A).
Abstract: It is known that every complex trace-zero matrix is the sum of four square-zero matrices, but not necessarily of three such matrices. In this note, we prove that for every trace-zero matrix A over an arbitrary field, there is a non-negative integer p such that the extended matrix is the sum of three square-zero matrices (more precisely, one can simply take p as the number of rows of A). Moreover, we demonstrate that if the underlying field has characteristic 2 then every trace-zero matrix is the sum of three square-zero matrices. We also discuss a counterpart of the latter result for sums of idempotents.
TL;DR: In this paper, a general algebraic approach to relationships between OLSEs and BLUEs of the whole and partial mean parameter vectors in a constrained general linear model (CGLM) with fixed parameters is provided.
Abstract: The well-known ordinary least-squares estimators (OLSEs) and the best linear unbiased estimators (BLUEs) under linear regression models can be represented by certain closed-form formulas composed by the given matrices and their generalized inverses in the models. This paper provides a general algebraic approach to relationships between OLSEs and BLUEs of the whole and partial mean parameter vectors in a constrained general linear model (CGLM) with fixed parameters by using a variety of matrix analysis tools on generalized inverses of matrices and matrix rank formulas. In particular, it establishes a variety of necessary and sufficient conditions for OLSEs to be BLUEs under a CGLM, which include many reasonable statistical interpretations on the equalities of OLSEs and BLUEs of parameter space in the CGLM. The whole work shows how to effectively establish matrix equalities composed by matrices and their generalized inverses and how to use them when characterizing performances of estimators of parameter spaces in linear models under most general assumptions.
01 Aug 2017
TL;DR: The blind source separation (BSS) problem is addressed, the optimal window length in the short-time Fourier transform (STFT) for independent low-rank matrix analysis (ILRMA) is analyzed, and the difference between the performances of ILRMA and conventional BSS is discussed.
Abstract: In this paper, we address the blind source separation (BSS) problem and analyze the optimal window length in the short-time Fourier transform (STFT) for independent low-rank matrix analysis (ILRMA). ILRMA is a state-of-the-art BSS technique that utilizes the statistical independence between low-rank matrix spectrogram models, which are estimated by nonnegative matrix factorization. In conventional frequency-domain BSS, the modeling error of a mixing system increases when the window length is too short, and the accuracy of statistical estimation decreases when the window length is too long. Therefore, the optimal window length is determined by both the reverberation time and the number of time frames. However, unlike classical BSS methods such as ICA and IVA, ILRMA enables the full modeling of spectrograms, which may improve the robustness to a decrease in the number of frames in a longer-window case. To confirm this hypothesis, the optimal window length for ILRMA is experimentally investigated, and the difference between the performances of ILRMA and conventional BSS is discussed.
01 Jul 2017
TL;DR: In this paper, a rank constraint on collections of fundamental matrices in multi-view settings was introduced to improve the recovery of camera locations, particularly in multiview settings in which fewer images are available.
Abstract: Accurate estimation of camera matrices is an important step in structure from motion algorithms. In this paper we introduce a novel rank constraint on collections of fundamental matrices in multi-view settings. We show that in general, with the selection of proper scale factors, a matrix formed by stacking fundamental matrices between pairs of images has rank 6. Moreover, this matrix forms the symmetric part of a rank 3 matrix whose factors relate directly to the corresponding camera matrices. We use this new characterization to produce better estimations of fundamental matrices by optimizing an L1-cost function using Iterative Re-weighted Least Squares and Alternate Direction Method of Multiplier. We further show that this procedure can improve the recovery of camera locations, particularly in multi-view settings in which fewer images are available.
TL;DR: A new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form.
01 Jan 2017
TL;DR: Different computational methods for generating invertible binary matrices with close to the maximum number of invertable 2 × 2 submatrices have been implemented and compared against each other and the ternary matricesWith such properties are studied.
Abstract: A linear 2-All-or-Nothing Transform can be considered as an invertible matrix with all 2 × 2 submatrices invertible. It is known [P. D’Arco, N. Nasr Esfahani and D.R. Stinson, Electron. J. Combin. 23(4) (2016), #P4.10] that there is no binary s×s matrix that satisfies these conditions, for s > 2. In this paper, different computational methods for generating invertible binary matrices with close to the maximum number of invertible 2 × 2 submatrices have been implemented and compared against each other. We also study the ternary matrices with such properties.
TL;DR: In this paper, the authors describe a systematic approach to bounding rates of convergence and proving tail inequalities for the empirical spectral measures of a wide variety of random matrix ensembles, including Wigner matrices, Wishart matrices and Haar-distributed matrices.
Abstract: Understanding the limiting behavior of eigenvalues of random matrices is the central problem of random matrix theory. Classical limit results are known for many models, and there has been significant recent progress in obtaining more quantitative, non-asymptotic results. In this paper, we describe a systematic approach to bounding rates of convergence and proving tail inequalities for the empirical spectral measures of a wide variety of random matrix ensembles. We illustrate the approach by proving asymptotically almost sure rates of convergence of the empirical spectral measure in the following ensembles: Wigner matrices, Wishart matrices, Haar-distributed matrices from the compact classical groups, powers of Haar matrices, randomized sums and random compressions of Hermitian matrices, a random matrix model for the Hamiltonians of quantum spin glasses, and finally the complex Ginibre ensemble. Many of the results appeared previously and are being collected and described here as illustrations of the general method; however, some details (particularly in the Wigner and Wishart cases) are new.
TL;DR: In this paper, it was shown that a map is additive if and only if for all rank-s matrices, where s is a fixed positive integer such that, which has been verified to be true for the case in [Xu X, Pei Y, Yi X. 2016;64:1283-1294].
Abstract: Let be the ring of all matrices over a field with . In this paper, it is proved that a map is additive if and only if for all rank-s matrices , where s is a fixed positive integer such that , which has been verified to be true for the case in [Xu X, Pei Y, Yi X. Additive maps on invertible matrices. Linear Multilinear Algebra. 2016;64:1283–1294]. Also, an example is provided showing that the conclusion will be incorrect if .
TL;DR: In this paper, a state-space matrix interpolation technique is proposed to generate linear parameter-varying (LPV) models starting from a set of local linear time-invariant (LTI) models estimated at fixed operating conditions.
TL;DR: The set of matrices also embed a covariance structure for the matrix coefficients and it is shown that when negative covariances predominate in the rows, more favorable optimal objective values for the primal can be expected.
Abstract: In this paper we consider robust linear programs with uncertainty sets defined by the convex hull of a finite number of m × n matrices. Embedded within the matrices are related robust linear programs defined by the rows, columns, and coefficients of the matrices. This results in a nested set of primal (and dual) linear programs with predictably different optimal objective values. The set of matrices also embed a covariance structure for the matrix coefficients and we show that when negative covariances predominate in the rows, more favorable optimal objective values for the primal can be expected.
TL;DR: In this article, the graph product rules combined with matrix analysis and linear algebra are used for static and dynamic analysis of near-regular mechanical systems, where a large number of degrees of freedom are affected by irregularity.
Abstract: Some near-regular mechanical systems involve global irregularities, wherein a large number of degrees of freedom are affected by irregularity. However, no efficient solution for such global near-regular systems has yet been developed. In this paper, methods for static and dynamic analyses/reanalyses of these systems are established using graph product rules combined with matrix analysis and linear algebra. Also, these methods are generalized to systems with nonlinear behavior. The developed formulations allow reduction in computational time and storage compared to those of conventional methods. As a practical example of a global near-regular mechanical system, a subject-specific finite element mechanical model of the human spine is developed and presented.
TL;DR: A family of symmetric matrices that are minimal for the Bruhat order on the class of 2k-by-2k ( 0 , 1 ) -matrices with constant row sums equal to k + 1 are described and identified.
TL;DR: In this paper, the authors show that if a linear space of symmetric matrices is made of nilpotents and when it is triangular, then the maximal dimension of L is n (n − 1 ) 2, and if this dimension is attained, then L is triangularizable.
TL;DR: It is proved that Toeplitz matrices are unitarily similar to complex symmetric matrices, and two unitary matrices that uniformly turn all Toe PL matrices via similarity to complex symmetry are explicitly given.
Abstract: We prove that Toeplitz matrices are unitarily similar to complex symmetric matrices Moreover, two unitary matrices that uniformly turn all Toeplitz matrices via similarity to complex symmetric mat
TL;DR: In this paper, a matrix analysis method is presented to deduce the analytical solution of Jacobian matrices of WDS models, including the nodal demand, pipe roughness and pipe diameter to nodal pressure and pipe flow, respectively.