Search

This paper considers the usual compound matrix, the additive compound matrix, and the discriminant matrix. An investigation of the Jordan structure of discriminant matrices is initiated. The primary objective of the paper is the determination of the Jordan form of the discriminant of one Jordan block. This Jordan form has a very simple pattern. A number of related results are obtained. The main results are facilitated by the new precedence relation ordering and the block-persymmetric property of the matrices. Examples of the discriminant matrices and Jordan forms are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new decomposition (called the core-EP decomposition in the present paper) for square matrices and give some of its applications. By applying the decomposition, we derive several characterizations of the core-EP inverse, introduce two orders – the core-EP order and the core-minus partial order, and characterize their properties. We also give some properties of the decomposition. [ABSTRACT FROM AUTHOR]

This paper explores M ( R ∞ 4 ) , the group of orientation preserving Möbius transformations acting in R ∞ 4 . On the one hand M ( R ∞ 4 ) is given by the group of 2 × 2 matrices over the quaternions H with determinant D derived from the corresponding 4 × 4 matrices over the complex numbers C . On the other hand we know that, in general, M ( R ∞ n ) may be given in terms of 2 × 2 matrices over the Clifford algebra C n with n − 1 generators. Thus when n = 4 , M ( R ∞ 4 ) is given in terms of 2 × 2 matrices with entries drawn from C 4 and determinant Δ defined in terms of the entries of the given matrix. We note that the skew field H may be considered as a Clifford algebra C 3 based on two generators i and j or more generally i 1 and i 2 where i j = k or i 1 i 2 = k , while the set of elements { 1 , i , j , k } form a basis of H regarded as a 4-dimensional real vector space. Thus H is embedded in C 4 . In the paper we reconcile the two representations of M ( R ∞ 4 ) by comparing the generating sets of the underlying groups of matrices. A relationship between a determinants D and Δ is also exposed. [ABSTRACT FROM AUTHOR]

This paper presents necessary and sufficient conditions for a square matrix over an arbitrary field to be expressed as a product of idempotent and square-zero matrices. Since the similarity structures of such factors are completely determined by their rank, we also investigate the ranks that these factors may have. The case where the factors are all idempotent or all square-zero have been investigated by various authors, some of whom are listed in the references of this paper. [ABSTRACT FROM AUTHOR]

We prove the determinant connectivity matrix formula. Mathematically, the proof introduces novel techniques based on an algebraic approach and connectivity properties. Although this is the second part of a previous paper and has its original motivation there, the paper is self contained and the result is interesting in itself. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we apply a complete pivoting strategy to compute the left-looking version of AINV preconditioner for linear systems. As the preprocessing, the MultiLevel Nested Dissection reordering has also been applied. We have used this preconditioner as the right preconditioner for several linear systems where the coefficient matrices have been downloaded from the University of Florida Sparse Matrix Collection. Numerical experiments presented in this paper indicate the effectiveness of such a complete pivoting on the quality of left-looking version of AINV preconditioner. [Copyright &y& Elsevier]

Abstract: This paper studies Symmetric Determinantal Representations (SDR) in characteristic 2, that is the representation of a multivariate polynomial P by a symmetric matrix M such that , and where each entry of M is either a constant or a variable. We first give some sufficient conditions for a polynomial to have an SDR. We then give a non-trivial necessary condition, which implies that some polynomials have no SDR, answering a question of Grenet et al. A large part of the paper is then devoted to the case of multilinear polynomials. We prove that the existence of an SDR for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings. We develop some algorithms to test the factorizability in these rings and use them to find SDRs when they exist. Altogether, this gives us polynomial-time algorithms to factorize the polynomials in the quotient rings and to build SDRs. We conclude by describing the case of Alternating Determinantal Representations in any characteristic. [Copyright &y& Elsevier]

Abstract: Let A be the adjacency matrix of a graph X and suppose . We view A as acting on and take the standard basis of this space to be the vectors for u in . hysicists say that we have perfect state transfer from vertex u to v at time if there is a scalar such that (Since is unitary, .) For example, if X is the d-cube and u and v are at distance d then we have perfect state transfer from u to v at time . Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from u to v if there is a complex number and, for each positive real there is a time t such that Again we necessarily have . In a recent paper Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path if and only is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for and .) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we extend the theory of pretty good state transfer. We provide what is only the second family of graphs where pretty good state transfer occurs. The graphs we use are the double-star graphs , these are trees with a vertex of degree adjacent to a vertex of degree , and all other vertices of degree one. We prove that perfect state transfer does not occur in any graph in this family. We show that if , then there is pretty good state transfer in between the two end vertices adjacent to the vertex of degree three. If , we prove that there is never perfect state transfer between the two vertices of degree at least three, and we show that there is pretty good state transfer between them if and only these vertices both have degree and is not a perfect square. Thus we find again the the existence of perfect state transfer depends on a number theoretic condition. It is also interesting that although no double stars have perfect state transfer, there are some that admit pretty good state transfer. [Copyright &y& Elsevier]

Abstract: In a recent paper [D. Zhang, Z. Lin, Y. Liu, On eigenvalues and equivalent transformation of trigonometric matrices, Linear Algebra Appl. 436 (2012) 71–78], the authors motivated and discussed a trigonometric matrix that arises in the design of finite impulse response (FIR) digital filters. The eigenvalues of this matrix shed light on the FIR filter design, so obtaining them in closed form was investigated. Zhang et al. proved that their matrix is rank-4 and they conjectured closed form expressions for its eigenvalues. This paper studies trigonometric matrices more general than theirs, deduces their rank, and derives closed-forms for their eigenvalues. As a corollary, it yields an elementary proof of the conjecture in the aforementioned paper. [Copyright &y& Elsevier]

Abstract: Given a Probabilistic Boolean Network (PBN), an important problem is to study its steady-state probability distribution for network analysis. In this paper, we present a new perturbation bound of the steady-state probability distribution of PBNs with gene perturbation. The main contribution of our results is that this new bound is established without additional condition required by the existing method. The other contribution of this paper is to propose a fast algorithm based on the special structure of a transition probability matrix of PBNs with gene perturbation to compute its steady-state probability distribution. Experimental results are given to demonstrate the effectiveness of the new bound, and the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity ? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5. [Copyright &y& Elsevier]

Abstract: In this paper, we slightly generalize the notion of G-matrices, which has been recently introduced. A real nonsingular matrix A is called a G-matrix if there exist nonsingular diagonal matrices and such that . We generalize this definition to the case where A can be singular. We say that a real matrix A, which is not necessarily square, is a generalized G-matrix (GG-matrix) if there exist nonsingular diagonal matrices and such that is a g-inverse of A. The main purpose of this paper is to show that any generalized Cauchy matrix is a GG-matrix. [Copyright &y& Elsevier]

Abstract: This paper extends Hua’s theorem on the geometry of rectangular matrices over a division ring to the case of Bezout domains. Let be integers , R an be two Bezout domains. Assume that is an adjacency preserving bijective map in both directions. Further, assume that is a local ring, or is an invertibility preserving map, or is an additive map. This paper obtains the algebraic formulas of . As applications, the ring semi-isomorphisms from to are characterized, and the group isomorphisms from to are discussed. [Copyright &y& Elsevier]

Abstract: Let be a solvable Lie subalgebra of the Lie algebra ( as a vector space). Let be polynomials in the commuting variables with coefficients in . For matrices , let and letIn this paper, we prove that, for , if one value of the matrix-valued function (the value depends on the product order of the variables) is nilpotent, then, (a) all values of are nilpotent; (b) all values of (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras. [Copyright &y& Elsevier]

Abstract: This paper investigates the relation between the eigenvalues of a Euclidean distance matrix (EDM) and those of the corresponding positive semidefinite matrix. More precisely, let and be two EDMs and let and be the corresponding positive semidefinite matrices such that . In this paper, when the eigenvalues of majorizes those of , a condition under which the eigenvalues of majorizes those of is derived. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a class of regular bipartite graphs whose biadjacency matrices are circulant matrices – a generalization of circulant graphs which happen to be bipartite – and we describe some of their properties. Notably, we compute upper and lower bounds for the zero forcing number for such a graph based only on the parameters that describe its biadjacency matrix. The main results of the paper characterize the bipartite circulant graphs that achieve equality in the lower bound and compute their minimum ranks. [Copyright &y& Elsevier]

Abstract: The paper consists of two parts. In the first part, we consider two matrices that appear in the literature in the study of irreducible Markov chains. The first matrix N is equal to the mean first passage of the Markov chain except on the diagonal where N vanishes. The other matrix K is equal to , where J is the all-1 matrix, A is the identity minus the transition matrix of the Markov chain, and is the diagonal matrix whose diagonal entries are the corresponding diagonal entries of the group inverse of A. Both N and K are known to be invertible. We show that the diagonal entries of and of are strictly negative in sufficiently high dimensions ( for N and for K). These results lead to a number of inequalities of independent interest, one of which we study in greater detail probabilistically. In the second part of the paper, we address a problem raised by Kemeny and Snell of determining whether a given Markov chain is primitive only from its first mean passage matrix, without having to compute the transition matrix. We derive several simple conditions of the mean first passage matrix which are helpful in determining whether the corresponding transition matrix is primitive. [Copyright &y& Elsevier]

Abstract: In this paper, we study the numerical behavior of the semismooth and smoothing Newton methods for solving Pareto eigenvalue problem of the formwhere (A, B) is a pair of possibly asymmetric matrices of order n. Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. By using the (smoothing) Fischer-Burmeister NCP function and the normalization condition e T x =1, the Pareto eigenvalue problem can be converted into a equivalent semismooth (or smoothing) system of equations, where e =(1,…,1) T . Then a semismooth (or smoothing) Newton algorithm is designed to solve such a semismooth (or smoothing) system of equations. Some numerical results are reported in the paper, which indicates that the proposed algorithms are very effective. [Copyright &y& Elsevier]

Abstract: Let G be a simple undirected graph with the characteristic polynomial of its Laplacian matrix , . Aleksandar Ilić [A. Ilić, Trees with minimal Laplacian coefficients, Comput. Math. Appl. 59 (2010) 2776–2783] identified n-vertex trees with given matching number q which simultaneously minimize all Laplacian coefficients. In this paper, we give another proof of this result. Generalizing the approach in the above paper, we determine n-vertex trees with given matching number q which have the second minimal Laplacian coefficients. We also identify the n-vertex trees with a perfect matching having the largest and the second largest Laplacian coefficients, respectively. Extremal values on some indices, such as Wiener index, modified hyper-Wiener index, Laplacian-like energy, incidence energy, of n-vertex trees with matching number q are obtained in this paper. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4. [Copyright &y& Elsevier]

Abstract: In an earlier paper the authors examined the problem of selecting rows of a matrix so that the resulting matrix is as “non-singular” as possible. However, the proof of the key result in that paper is not constructive. In this note we give a constructive proof for that result. In addition, we examine a case where as non-singular as possible means maximizing a determinant and provide a new bound and a constructive proof for this case also. [Copyright &y& Elsevier]

Abstract: In a recent paper [7], Gowda et al. extended Ostrowski–Schneider type inertia results to certain linear transformations on Euclidean Jordan algebras. In particular, they showed that whenever by the min–max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined bywith , , and denoting, respectively, the number of positive, negative, and zero eigenvalues, counting multiplicities. In this paper, we present a Peirce decomposition version of Wimmer’s result [13] and show that it is equivalent to the above result. In addition, we extend Higham and Cheng’s result ([8], Lemma 4.2) to the setting of Euclidean Jordan algebras. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject. As a corollary we get the main results from the paper of Yang and Liu [H. Yang, X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math. 235 (2011) 1412–1417]. As an application we give some new representations for the Drazin inverse of a block matrix. [Copyright &y& Elsevier]

Abstract: Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results. Previous work has focused on characterizing -quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between -quasiseparable matrices and the subclass of -quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices. [Copyright &y& Elsevier]

Abstract: Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results. Previous work has focused on characterizing -quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between -quasiseparable matrices and the subclass of -quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices. [Copyright &y& Elsevier]

This paper addresses the frequency response of coupled bending-torsional beams carrying an arbitrary number of in-span viscoelastic dampers and attached masses. Using the elementary coupled bending-torsion theory, along with appropriate generalized functions to treat the discontinuities of the response variables at the application points of dampers/masses, exact analytical expressions are derived for the frequency response of the beam under harmonically-varying, arbitrarily-placed point/polynomial loads. On this basis, the exact 6 × 6 dynamic stiffness matrix and 6 × 1 load vector of a two-node coupled bending-torsional beam finite element, with any number of in-span dampers/masses and harmonic loads, are obtained in a closed analytical form. Finally, the modal frequency response functions of the beam are built by a complex modal analysis approach, upon deriving pertinent orthogonality conditions for the modes. In this context, the modal impulse response functions are also obtained for time-domain analysis under arbitrary loads. [ABSTRACT FROM AUTHOR]

The paper examines the problem of wave propagation in a random conducting magneto- non-simple thermo-viscoelastic medium. The medium has been assumed to be weakly conducting and weakly thermal. The thermomechanical coupling parameter and the conductivity are random functions, proportional to ε, with non-zero mean values, ε measuring the smallness of the scale of random fluctuation of inhomogeneities of the medium. The smooth perturbation technique enunciated by Keller (1964) has been employed to analyze the appropriate dispersion equation in non-simple thermoelastic medium. The longitudinal and transverse waves were discussed by using a particular form of thermomechanical coupling parameter representing the corresponding auto-correlation function. The effect of magnetic conductivity has been investigated. The phenomena of attenuation of waves and change of phase speed were discussed numerically in details. [ABSTRACT FROM AUTHOR]

While classical matrix and finite-element methods are very powerful for structural analysis, some essential shortcomings such as not placing the transverse and axial loads in their actual locations on a member, as well as necessity of a large number of iterations or subdivisions of the structural element for exact second-order analysis should be still taken into consideration. To overcome such drawbacks, Generalized Differential Quadrature (GDQ) is a new semi-numerical method for analysis of the engineering problems. The aim of this paper is to use SBCGS (Satisfaction of Boundary Conditions in the General Solution) technique to perform second-order and buckling analysis of beam-columns under different geometrical and boundary conditions by solving beam- column differential equation in matrix form. In addition, a comparison is made between the results of SBCGS technique and those of other methods available in literature for the analysis of non-prismatic beam-columns with different tapered ratios when subjected to end or distributed axial loads. Using the proposed method, a rigorous second-order analysis on a beam-column member with variable cross-section and various boundary conditions under different axial and transverse distributed loads can be carried out just by one iteration and without any discretization in the member but instability appears in the results when axial load reaches critical buckling load. Moreover, in order to achieve the accuracy of the results of SBCGS technique, the model must have a large number of subdivisions in FEM method. [ABSTRACT FROM AUTHOR]

In this paper, we give a new angle to interpret Riordan arrays by showing that every Riordan array can be expressed as a Krylov matrix. We then use this idea to obtain some groups containing the Riordan group as a subgroup. Moreover, we study Lie algebras for the extended Riordan groups as Lie groups. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce new necessary and sufficient conditions for an Euclidean distance matrix to be multispherical. The class of multispherical distance matrices studied in this paper contains not only most of the matrices studied by Hayden et al. (1996) , but also many other multispherical structures that do not satisfy the conditions in Hayden et al. (1996) . We also study the information provided by the origin of coordinates when it is placed at the center of the spheres and the origin representation property is satisfied. These vectors associated with the origin of coordinates generate a number of supporting hyperplanes for a family of multispherical matrices and also describe part of the null space of the corresponding distance matrices. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the numerical solution of large scale Sylvester equations , Lyapunov equations as a special case in particular included, with having very small rank. For stable Lyapunov equations, Penzl (2000) and Li and White (2002) demonstrated that the so-called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. An easily implementable extension of Penz’s shift strategy for the Lyapunov equation is presented for the current case. It is demonstrated that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions. [Copyright &y& Elsevier]

Abstract: Let be the real quaternion algebra and denote the set of all matrices over . Let and be involutions, i.e., . A matrix is said to be -symmetric if . This paper studies the system of linear real quaternion matrix equationsWe present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied. As applications, we discuss the necessary and sufficient conditions for the systemto have a -symmetric solution. We also show an expression of the -symmetric solution to the system when the solvability conditions are met. Moreover, we provide an algorithm and a numerical example to illustrate our results. The findings of this paper extend some known results in the literature. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix , a diagonal matrix such that all elements of are less than or equal to the maximum cycle geometric mean of , with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry. [Copyright &y& Elsevier]

Abstract: Let and be respectively the adjacency matrix and the degree matrix of a graph . The signless Laplacian matrix of is defined as . The -spectrum of is the set of the eigenvalues together with their multiplicities of . The -index of is the maximum eigenvalue of . The possibilities for developing a spectral theory of graphs based on the signless Laplacian matrices were discussed by Cvetković et al. [D. Cvetković, P. Rowlinson, S.K. Simić, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155–171]. In the latter paper the authors determine the graphs whose -index is in the interval . In this paper, we investigate some properties of -spectra of graphs, especially for the limit points of the -index. By using these results, we characterize respectively the structures of graphs whose the -index lies in the intervals , and , where . [Copyright &y& Elsevier]

Abstract: In this paper, we study potential convergence of modulus patterns. A modulus pattern is convergent if all complex matrices in (i.e. all matrices with modulus pattern ) are convergent. A modulus pattern is potentially (absolutely) convergent if there exists a (nonnegative) convergent matrix in . We also introduce types of potential convergence that correspond to diagonal and D-convergence, studied in [E. Kaszkurewicz, A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhauser, 2000]. Convergent modulus patterns have been completely characterized by Kaszkurewicz and Bhaya [E. Kaszkurewicz, A. Bhaya, Qualitative stability of discrete-time systems, Linear Algebra Appl. 117 (1989) 65–71]. This paper presents some techniques that can be used to establish potential convergence. Potential absolute convergence and potential diagonal convergence are shown to be equivalent, and their complete characterization for modulus patterns is given. Complete characterizations of all introduced types of potential convergence for modulus patterns are also presented. [Copyright &y& Elsevier]

Abstract: The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized -matrices. A truth is found that the class of conjugate generalized -matrices is a subclass of the class of generalized -matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized -matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized -matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper. [Copyright &y& Elsevier]

Abstract: For linear descriptor systems of the form , , this paper constructs reduced order systems associated with a given part of the finite spectrum of the pencil . It is known that the reduction can be obtained by a block diagonalization of the generalized Schur decomposition of . In this paper we consider the special case when and . This case is suited, in particular, for linearized hydrodynamic problems. We derive a sufficient condition under which the reduced system can approximate the initial one and show that it can be obtained in significantly cheap and efficient approaches. We consider first in detail the case when and is the identity matrix and then treat the general case. [Copyright &y& Elsevier]

Abstract: Threshold group testing first proposed by Damaschke is a generalization of classic group testing. Specifically, a group test is positive (negative) if it contains at least (at most ) positives, and if the number of positives is between and , the test outcome is arbitrary. Although sequential group testing algorithms have been proposed, it is unknown whether an efficient nonadaptive algorithm exists. In this paper, we give an affirmative answer to this problem by providing efficient nonadaptive algorithms for the threshold model. The key observation is that disjunct matrices, a standard tool for group testing designs, also work in this threshold model. This paper improves and extends previous results in three ways: 1. The algorithms we propose work in one stage, which saves time for testing. 2. The test complexity is lower than previous results, at least for the number of elements which need to be tested is sufficiently large. 3. A limited number of erroneous test outcomes are allowed. [Copyright &y& Elsevier]

Abstract: This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms. [Copyright &y& Elsevier]

Abstract: Let and f be a multiplicative function such that for every positive integer k and the Euler product converges. Let be the matrix defined on the set C having f evaluated at the greatest common divisor of i and j as its ij-entry. In the present paper, we first obtain the least upper bounds for the ij-entry and the absolute row sum of any row of , the inverse of , in terms of . Specializing these bounds for the arithmetical functions and we examine the asymptotic behavior the smallest eigenvalue of each of matrices and depending on when m tends to infinity. We conclude our paper with a proof of a conjecture posed by Hong and Loewy [S. Hong, R. Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasg. Math. J. 46 (2004) 551–569]. [Copyright &y& Elsevier]

Abstract: Tucker three-way PCA and Candecomp/Parafac are two well-known methods of generalizing principal component analysis to three way data. Candecomp/Parafac yields component matrices (e.g., for subjects or objects), (e.g., for variables) and (e.g., for occasions) that are typically unique up to jointly permuting and rescaling columns. Tucker-3 analysis, on the other hand, has full transformational freedom. That is, the fit does not change when , and are postmultiplied by nonsingular transformation matrices, provided that the inverse transformations are applied to the so-called core array . This freedom of transformation can be used to create a simple structure in , and/or in . This paper deals with the latter possibility exclusively. It revolves around the question of how a core array, or, in fact, any three-way array can be transformed to have a maximum number of zero elements. Direct applications are in Tucker-3 analysis, where simplicity of the core may facilitate the interpretation of a Tucker-3 solution, and in constrained Tucker-3 analysis, where hypotheses involving sparse cores are taken into account. In the latter cases, it is important to know what degree of sparseness can be attained as a tautology, by using the transformational freedom. In addition, simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three-way arrays. So far, a number of simplicity results have been attained, pertaining to arrays sampled randomly from continuous distributions. These results do not apply to three-way arrays with symmetric slices in one direction. The present paper offers a number of simplicity results for arrays with symmetric slices of order and . Some generalizations to higher orders are also discussed. As a mathematical application, the problem of determining the typical rank of and arrays with symmetric slices will be revisited, using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one, that can be attained almost surely for such arrays. The issue of maximal simplicity of the targets to be presented will be addressed, either by formal proofs or by relying on simulation results. [Copyright &y& Elsevier]

Abstract: The principal result in this paper is concerned with the derivative of a vector with respect to a block vector or matrix. This is applied to the asymptotic Fisher information matrix (FIM) of a stationary vector autoregressive and moving average time series process (VARMA). Representations which can be used for computing the components of the FIM are then obtained. In a related paper [A. Klein, A generalization of Whittle’s formula for the information matrix of vector mixed time series, Linear Algebra Appl. 321 (2000) 197–208], the derivative is taken with respect to a vector. This is obtained by vectorizing the appropriate matrix products whereas in this paper the corresponding matrix products are left unchanged. [Copyright &y& Elsevier]

Abstract: The concept of a hypergeneralized projector as a matrix H satisfying H 2 = H †, where H † denotes the Moore–Penrose inverse of H, was introduced by Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. In the present paper, the problem of when a linear combination c 1 H 1 + c 2 H 2 of two hypergeneralized projectors H 1, H 2 is also a hypergeneralized projector is considered. Although, a complete solution to this problem remains unknown, this article provides characterizations of situations in which (c 1 H 1 + c 2 H 2)2 =(c 1 H 1 + c 2 H 2)† derived under certain commutativity property imposed on matrices H 1 and H 2. The results obtained substantially generalize those given in the above mentioned paper by Groß and Trenkler. [Copyright &y& Elsevier]

Abstract: In a recent paper [C.R. Johnson, S. Furtado, A generalization of Sylvester’s law of inertia, Linear Algebra Appl. 338 (2001) 287–290], Sylvester’s law of inertia is generalized to any matrix that is ∗-congruent to a diagonal matrix. Such a matrix is called unitoid. In the present paper, an alternative approach to the subject of unitoidness is offered. Specifically, Sylvester’s law of inertia states that a Hermitian n × n matrix of rank r with inertia (p, q, n − r) is ∗-congruent to the direct sumIt is demonstrated herein that a unitoid matrix A of rank r is ∗-congruent to a direct sum of diagonal blocks of the formtogether with the zero block 0I n−r . Moreover, the ϕ’s together with the multiplicities p and q are specified in terms of the eigenvalues and eigenvectors of A † A ∗, where A † is the Moore–Penrose inverse of A. [Copyright &y& Elsevier]

Abstract: Although principal component analysis (PCA) is an important tool in standard multivariate data analysis, little interest has been devoted to assessing whether the underlying relationship within a given variable set can be described by a linear PCA model or whether nonlinear PCA must be utilized. This paper addresses this deficiency by introducing a nonlinearity measure for principal component models. The measure is based on the following two principles: (i) the range of recorded process operation is divided into smaller regions; and (ii) accuracy bounds are determined for the sum of the discarded eigenvalues. If this sum is within the accuracy bounds for each region, the process is assumed to be linear and vice versa. This procedure is automated through the use of cross-validation. Finally, the paper shows the utility of the new nonlinearity measure using two simulation studies and with data from an industrial melter process. [Copyright &y& Elsevier]

This paper deals with several ways to generalize the Flanders’ theorem to matrix triples. We consider six invertible matrices and try to write them as the possible products of three matrices. Initially, we describe a wide set of necessary conditions so that this system be solvable, showing that they are not sufficient. Next, we study the simultaneous solvability of two equations, selected appropriately among the matrix system. The rest of the paper is devoted to the study of a particular case, in which the six given matrices are simultaneously diagonalizable, with distinct nonzero eigenvalues. In this case, we obtain a necessary and sufficient condition for the solvability of the full matrix system. Moreover, an explicit solution to it is constructed. Certain technical results necessary for this work may be of independent interest. [Copyright &y& Elsevier]

In earlier papers, the Bauer–Fike technique was applied to the ordinary eigenvalue problem Ax=λx, the generalized eigenvalue problem Ax=λBx and the matrix polynomial eigenvalue problem ∑k=0mλkAkx=0. General multiple eigenvalues were dealt with and condition numbers were obtained for individual as well as clusters of eigenvalues. In this paper, we shall generalize the technique to the eigenvalue problem for regular periodic matrix pairs. Perturbation to eigenvalues, simple or multiple, finite or infinite, will be considered for perturbations of any size. For small perturbations, clusters of eigenvalues will be considered. [Copyright &y& Elsevier]

Using sequence information can improve performances in fault detection for serial (temporal) correlated process data. Classical methods firstly construct extended vectors through concatenating current process data and a certain number of previous process data, and then take dimension reduction methods. However, the simple extension of process data may distort the correlation between variables and largely increase the dimensionality. This paper proposes a novel algorithm, called Dynamic Graph Embedding (DGE), for fault detection. DGE adopts augmented matrices instead of extended vectors to encode sequence information. Furthermore, DGE incorporates both time information and neighborhood information to form similarities of different process data. And then DGE is designed to obtain embedding matrices with Markov chain analysis of the similarities. Extensive experimental results on the Tennessee Eastman (TE) benchmark process show the superiority of DGE in terms of missed detection rate (MDR) and false alarm rate (FAR). [ABSTRACT FROM AUTHOR]

Let U T n ( F ) be the algebra of n × n upper-triangular matrices over an algebraically closed field F of characteristic zero. In [18] , the authors described all abelian G -gradings on U T n ( F ) by showing that any G -grading on this algebra is an elementary grading. In this paper, we shall consider the algebra U T n ( F ) endowed with an elementary Z 2 -grading. In this way, it has a structure of superalgebra and our goal is to completely describe the superinvolutions which can be defined on it. To this end, we shall prove that the superinvolutions and the graded involutions (i.e., involutions preserving the grading) on U T n ( F ) are strictly related through the so-called superautomorphisms of this algebra. We shall show that there exist two classes of inequivalent superinvolutions when n is even and a single class otherwise. Along the way, we shall give a complete description of the polynomial identities and the cocharacter sequences of U T 2 ( F ) and U T 3 ( F ) endowed with all possible superinvolutions. [ABSTRACT FROM AUTHOR]

Let k , n denote two positive integers and consider the family of the polytopes defined as the convex hull of pairs of the form ( Y , h ) where Y is a 0/1-matrix with k rows, n columns, containing exactly one nonzero coefficient per column, and where h stands for the smallest index of a nonzero row of Y . These polytopes and some variants naturally emerge in formulations of different classical combinatorial optimization problems such as minimum makespan scheduling and minimum span frequency assignment. In this paper, we provide complete formulations for these polytopes and show the associated separation problem can be solved in polynomial time. The complete formulations in the original space of variables generally contain an exponential number of inequalities. Alternative extended compact formulations are also presented. [ABSTRACT FROM AUTHOR]