Certain Class Of Matrices Research Articles

Matrices with hyperbolical Krein space numerical range

AbstractThis paper is devoted to matrices with hyperbolical Krein space numerical range. This shape characterizes the 2-by-2 case and persists for certain classes of matrices, independently of their size. Necessary and sufficient conditions for low dimensional tridiagonal matrices to have this shape are obtained involving only the matrix entries.

On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175 (2) (2019) 384–401, MR3968860].

Solitons of the vector KdV and Yamilov lattices

We study a vector generalizations of the lattice KdV equation and one of the simplest Yamilov equations. We use algebraic properties of a certain class of matrices to derive the N-soliton solutions.

Methods for Nonnegative Matrix Factorization Based on Low-Rank Cross Approximations

Available methods for nonnegative matrix factorization make use of all elements of the original $$m \times n$$ matrix, and their complexity is at least $$O(mn),$$ which makes them extremely resource-intensive in the case of large amounts of data. Accordingly, the following natural question arises: given the nonnegative rank of a matrix, can a nonnegative matrix factorization be constructed using some of its rows and columns? Methods for solving this problem are proposed for certain classes of matrices, namely, for nonnegative separable matrices (for which there exists a cone spanned by several columns of the original matrix that contains all its columns), for nonnegative separable matrices with perturbations, and for nonnegative matrices of rank 2. In practice, the number of operations and the amount of storage used by the proposed algorithms depend linearly on $$m + n$$ .

A Monte Carlo method for computing the action of a matrix exponential on a vector

A Monte Carlo method for computing the action of a matrix exponential for a certain class of matrices on a vector is proposed. The method is based on generating random paths, which evolve through the indices of the matrix, governed by a given continuous-time Markov chain. The vector solution is computed probabilistically by averaging over a suitable multiplicative functional. This representation extends the existing linear algebra Monte Carlo-based methods, and was used in practice to develop an efficient algorithm capable of computing both, a single entry or the full vector solution. Finally, several relevant benchmarks were executed to assess the performance of the algorithm. A comparison with the results obtained with a Krylov-based method shows the remarkable performance of the algorithm for solving large-scale problems.

Nearness results for real tridiagonal 2‐Toeplitz matrices

SummaryIn this article, we extend the results for Toeplitz matrices obtained by Noschese, Pasquini, and Reichel. We study the distance d, measured in the Frobenius norm, of a real tridiagonal 2‐Toeplitz matrix T to the closure of the set formed by the real irreducible tridiagonal normal matrices. The matrices in , whose distance to T is d, are characterized, and the location of their eigenvalues is shown to be in a region determined by the field of values of the operator associated with T, when T is in a certain class of matrices that contains the Toeplitz matrices. When T has an odd dimension, the eigenvalues of the closest matrices to T in are explicitly described. Finally, a measure of nonnormality of T is studied for a certain class of matrices T. The theoretical results are illustrated with examples. In addition, known results in the literature for the case in which T is a Toeplitz matrix are recovered.

Optimizing Gershgorin for symmetric matrices

The Gershgorin Circle Theorem is a well-known and efficient method for bounding the eigenvalues of a matrix in terms of its entries. If A is a symmetric matrix, by writing A=B+x1, where 1 is the matrix with unit entries, we consider the problem of choosing x to give the optimal Gershgorin bound on the eigenvalues of B, which then leads to one-sided bounds on the eigenvalues of A. We show that this x can be found by an efficient linear program (whose solution can in may cases be written in closed form), and we show that for large classes of matrices, this shifting method beats all existing piecewise linear or quadratic bounds on the eigenvalues. We also apply this shifting paradigm to some nonlinear estimators and show that under certain symmetries this also gives rise to a tractable linear program. As an application, we give a novel bound on the lowest eigenvalue of a adjacency matrix in terms of the “top two” or “bottom two” degrees of the corresponding graph, and study the efficacy of this method in obtaining sharp eigenvalue estimates for certain classes of matrices.

TAKAGİ-SUGENO BULANIK COHEN-GROSSBERG TİPİ ZAMAN GECİKMELİ YAPAY SİNİR AĞLARINDA KARARLILIK ANALİZİ

This paper deals with the problem of the global asymptotic stability of the class of Takagi-Sugeno Fuzzy Cohen-Grossberg neural networks with multiple time delays. By constructing a suitable fuzzy Lyapunov functional, we present a new delay-independent sufficient condition for the global asymptotic stability of the equilibrium point for delayed Takagi-Sugeno Fuzzy Cohen-Grossberg neural networks with respect to the Lipschitz activation functions. The obtained condition simply relies on the network parameters of the neural system. Therefore, the equilibrium and stability properties of the neural network model considered in this paper can be easily verified by exploiting some basic properties of some certain classes of matrices.

The method of similar operators in the study of the spectra of the adjacency matrices of graphs

The method of similar operators [1, 2, 3] is used to investigate spectral properties of a certain class of matrices in the context of graphs [4, 5]. Specifically, we consider the adjacency matrix of an “almost-complete graph”. Then we generalize the result to allow the matrices obtained as combinations of the Kronecker products [6, 7] and the small-norm perturbations. We derive the estimates of the spectra and the eigenvectors of such matrices.

Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization

Given a matrix $M$ (not necessarily nonnegative) and a factorization rank $r$, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix $U$ with $r$ columns and a nonnegative matrix $V$ with $r$ rows such that $UV$ is the best possible approximation of $M$ according to some metric. In this paper, we study the properties of semi-NMF from which we develop exact and heuristic algorithms. Our contribution is threefold. First, we prove that the error of a semi-NMF of rank $r$ has to be smaller than the best unconstrained approximation of rank $r-1$. This leads us to a new initialization procedure based on the singular value decomposition (SVD) with a guarantee on the quality of the approximation. Second, we propose an exact algorithm (that is, an algorithm that finds an optimal solution), also based on the SVD, for a certain class of matrices (including nonnegative irreducible matrices) from which we derive an initialization for matrices not belonging to that class. Numerical experiments illustrate that this second approach performs extremely well, and allows us to compute optimal semi-NMF decompositions in many situations. Finally, we analyze the computational complexity of semi-NMF proving its NP-hardness, already in the rank-one case (that is, for $r = 1$), and we show that semi-NMF is sometimes ill-posed (that is, an optimal solution does not exist).

Generalization of some results concerning eigenvalues of a certain class of matrices and some applications

In this note, we present a generalization of some results concerning the spectral properties of a certain class of block matrices. As applications, we study some of its implications on nonnegative matrices and doubly stochastic matrices as well as on graph spectra and graph energy.

Estimation of bounds for the zeros of a polynomial using numerical radius

The estimation of zeros of a polynomial have been done by many mathematicians over the years using various approaches. In this paper we estimate the upper bound for the zeros of a given polynomial using Hilbert space technique involving Frobenius companion matrix and numerical radius. We first obtain numerical range and numerical radius for certain class of matrices and use them to estimate the bounds for zeros of a given polynomial. We illustrate with examples to show that the estimations obtained here is better than the previously known estimations. We also obtain a sequence of real numbers which converges exactly to the spectral radius of some special class of matrices.

SOME REMARKS ON THE DOZIER–SILVERSTEIN THEOREM FOR RANDOM MATRICES WITH DEPENDENT ENTRIES

The Dozier–Silverstein theorem asserts the almost sure convergence of the empirical spectral distribution of information plus noise matrices, i.e. perturbations of deterministic matrices whose spectral distribution converges (information matrices) by random matrices with i.i.d. entries (noise matrices). We show that a modification of the original proof given by Dozier and Silverstein allows to extend this result to more general noise matrices, in particular matrices with independent columns satisfying a natural concentration inequality for quadratic forms, matrices with independent entries, satisfying a Lindeberg-type condition (recovering a recent result by Xie), matrices with heavy-tailed entries in the domain of attraction of the Gaussian distribution and certain classes of matrices with dependencies among columns (generalizing those investigated recently by O'Rourke and containing matrices with exchangeable entries). As a corollary we obtain the circular law for random matrices with independent log-concave isotropic rows.

Accurate and efficient LDU decomposition of diagonally dominant M-matrices

NA † Abstract. An efficient method for the computation to high relative accuracy of the LDU decomposition of an n × n row diagonally dominant M-matrix is presented, assuming that the off-diagonal entries and row sums are given. This method costs an additional O(n2) elementary operations over the cost of Gaussian elimination, and leads to a lower triangular, column diagonally dominant matrix and an upper triangular, row diagonally dominant matrix. Comparisons with other methods in the literature are commented and illustrated. 1. Introduction. Recent advances in Numerical Linear Algebra have shown that certain classes of matrices allow computation of certain matrix functions to high relative accuracy, independently of the size of the classical condition number. Some of these classes of matrices are defined by special sign or other structure and require knowledge of some natural parameters to high relative accuracy. In most of those cases, accurate spectral computation (eigenvalues, singular values) is assured once we have an accurate matrix factorization with a suitable pivoting. For instance, the bidiagonal decomposition in the case of totally nonnegative matrices (see also (5), (10)) or an LDU factorization after a symmetric pivoting in the case of diagonally dominant matrices (cf. (4), (13), (14)). Let us focus now on the problem considered in this paper. An algorithm published in (2) computes with high relative accuracy the LDU factorization of an n × n row diagonally dominant M-matrix, if the off-diagonal entries and the row sums are given. The trick is to modify Gaussian elimination to compute the off-diagonal entries and the row sums of each Schur complement without performing subtractions. In addition, symmetric complete pivoting was used in (4) in order to obtain well conditioned L and U factors (U is even row diagonally dominant). This factorization is a special case of a

Maximal determinant over a certain class of matrices and its application to D-optimality of designs

It is known that some optimality criteria of experimental designs are functionals of the eigenvalues of their information matrices. In this context we study the problem of maximizing the determinant of α I t - ( P + P T ) , α > 2 , over the class of t-by- t permutation matrices, and the determinant of α I t + P + P T , α ≥ 2.5 , over the class of t-by- t permutation matrices with zero diagonal (derangement matrices). The results are used to characterize D-optimal complete block designs under an interference model.

Lattice Reduction Aided MMSE Decision Feedback Equalizers

The LLL algorithm developed by Lenstra, Lenstra, and Lovász has been widely studied to improve the conditioning of linear systems of equations that arise in multi-input multi-output (MIMO) antenna systems. In the first part of this correspondence, we show how a certain class of matrices can be transformed so that the LLL algorithm can be carried out efficiently, without compromising its performance. Then, the proposed transformation is applied to the matrix with block lower triangular Toeplitz matrices that arises in the decision feeedback equalizer (DFE) to improve the bit error rate (BER) performance with a slight computational overhead.

On Chebyshev Polynomials of Matrices

The $m$th Chebyshev polynomial of a square matrix $A$ is the monic polynomial that minimizes the matrix 2-norm of $p(A)$ over all monic polynomials $p(z)$ of degree $m$. This polynomial is uniquely defined if $m$ is less than the degree of the minimal polynomial of $A$. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well-known properties of Chebyshev polynomials of compact sets in the complex plane. We also derive explicit formulas of the Chebyshev polynomials of certain classes of matrices, and explore the relation between Chebyshev polynomials of one of these matrix classes and Chebyshev polynomials of lemniscatic regions in the complex plane.

Mathematical and computational tools in theoretical chemistry

It is with pleasure that we offer the readers of the Journal of Mathematical Chemistry, JCM, this Special Issue consisting of the contributions to Mathematical and computational tools in theoretical chemistry presented at the 8th International Conference of Computational and Mathematical Methods in Science and Engineering, CMMSE2008, held at Hotel Melia Galua, La Manga del Mar Menor, Murcia, Spain, June 13–16, 2008. This special issue is another one in a line of CMMSE special issues, prior advances are in the references [1–5]. As the main emphasis and profile of each CMMSE conference state, viz. they are multi-disciplinary, they combine important and novel mathematical and computational methods as well as innovative modelling and algorithms common to a wide spectrum of areas of human activity and finally new insights and understanding of the necessary development and evolution of present state of the art of mathematical and computational techniques and models. In this particular issue we will find, as the first article, important improvements in convergence and stability of applications to birth and death processes in chemical phenomena via iterative refinement of a certain class of matrices. A completely different contribution concerning neutron rich matter then follows. Since the methods, used here by the author, have a large overlap with the standard machinery of Quantum Chemistry, for instance the interaction model, the spherical harmonics analysis, and further with general techniques used in Theoretical Chemistry and Chemical Physics, e.g. cluster simulations, thermodynamic dependencies etc., the editors found the

Analysis of growth curve data by using cubic smoothing splines

Longitudinal data frequently arises in various fields of applied sciences where individuals are measured according to some ordered variable, e.g. time. A common approach used to model such data is based on the mixed models for repeated measures. This model provides an eminently flexible approach to modeling of a wide range of mean and covariance structures. However, such models are forced into a rigidly defined class of mathematical formulas which may not be well supported by the data within the whole sequence of observations. A possible non-parametric alternative is a cubic smoothing spline, which is highly flexible and has useful smoothing properties. It can be shown that under normality assumption, the solution of the penalized log-likelihood equation is the cubic smoothing spline, and this solution can be further expressed as a solution of the linear mixed model. It is shown here how cubic smoothing splines can be easily used in the analysis of complete and balanced data. Analysis can be greatly simplified by using the unweighted estimator studied in the paper. It is shown that if the covariance structure of random errors belong to certain class of matrices, the unweighted estimator is the solution to the penalized log-likelihood function. This result is new in smoothing spline context and it is not only confined to growth curve settings. The connection to mixed models is used in developing a rough testing of group profiles. Numerical examples are presented to illustrate the techniques proposed.

Extensions of Certain Graph-based Algorithms for Preconditioning

The original TPABLO algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries. This block diagonal can then be used as a preconditioner. We propose and analyze three extensions of this approach: We incorporate a nonsymmetric permutation to obtain a large diagonal, we use a more general parametrization for TPABLO, and we use a block Gauss–Seidel preconditioner which can be implemented to have the same execution time as the corresponding block Jacobi preconditioner. Experiments are presented showing that for certain classes of matrices, the block Gauss–Seidel preconditioner used with the system permuted with the new algorithm can outperform the best ILUT preconditioners in a large set of experiments.