Error-free matrix symmetrizers and equivalent symmetric matrices

A symmetrizer of the matrix A is a symmetric solution X that satisfies the matrix equation XA=AprimeX. An exact matrix symmetrizer is computed by obtaining a general algorithm and superimposing a modified multiple modulus residue arithmetic on this algorithm. A procedure based on computing a symmetrizer to obtain a symmetric matrix, called here an equivalent symmetric matrix, whose eigenvalues are the same as those of a given real nonsymmetric matrix is presented.

A parallel QR-algorithm for tridiagonal symmetric matrices /

"NSF-OCA-GJ-36936-000008."

A lower bound for the energy of symmetric matrices and graphs

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound $2\sqrt{m}$, where $m$ is the number of edges of the graph.

Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms

Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices under *congruence are given over an algebraically closed field of characteristic different from 2. (C) 2008 Elsevier Inc. All rights reserved.

On computing an equivalent symmetric matrix for a nonsymmetric matrix

A real or a complex symmetric matrix is defined here as an equivalent symmetric matrix for a real nonsymmetric matrix if both have the same eigenvalues. An equivalent symmetric matrix is useful in computing the eigenvalues of a real nonsymmetric matrix. A procedure to compute equivalent symmetric matrices and its mathematical foundation are presented.

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

A note on the eigenvalues of a special class of matrices

In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1, 1] for all values of m (the order of the matrix) and all values of a positive parameter a, the stability parameter sigma. As the order of the matrix is general, and the parameter sigma lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices. (C) 2010 Elsevier B.V. All rights reserved.

The discriminants associated to isotropy representations of symmetric spaces

We consider a generalized discriminant associated to a symmetric space which generalizes the discriminant of real symmetric matrices, and note that it can be written as a sum of squares of real polynomials. A method to estimate the minimum number of squares required to represent the discrimininant is developed and applied in examples.

Dictionary learning and sparse coding on Grassmann manifolds : an extrinsic solution

Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

An eigenproblem approach to classical screw theory

This paper presents a novel algebraic formulation of the central problem of screw theory, namely the determination of the principal screws of a given system. Using the algebra of dual numbers, it shows that the principal screws can be determined via the solution of a generalised eigenproblem of two real, symmetric matrices. This approach allows the study of the principal screws of the general two-, three-systems associated with a manipulator of arbitrary geometry in terms of closed-form expressions of its architecture and configuration parameters. We also present novel methods for the determination of the principal screws for four-, five-systems which do not require the explicit computation of the reciprocal systems. Principal screws of the systems of different orders are identified from one uniform criterion, namely that the pitches of the principal screws are the extreme values of the pitch.The classical results of screw theory, namely the equations for the cylindroid and the pitch-hyperboloid associated with the two-and three-systems, respectively have been derived within the proposed framework. Algebraic conditions have been derived for some of the special screw systems. The formulation is also illustrated with several examples including two spatial manipulators of serial and parallel architecture, respectively.

An optimal parallel algorithm for sparse matrix bandwidth reduction on a linear array

A simple and efficient algorithm for the bandwidth reduction of sparse symmetric matrices is proposed. It involves column-row permutations and is well-suited to map onto the linear array topology of the SIMD architectures. The efficiency of the algorithm is compared with the other existing algorithms. The interconnectivity and the memory requirement of the linear array are discussed and the complexity of its layout area is derived. The parallel version of the algorithm mapped onto the linear array is then introduced and is explained with the help of an example. The optimality of the parallel algorithm is proved by deriving the time complexities of the algorithm on a single processor and the linear array.

Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams

We present a complete solution to the problem of coherent-mode decomposition of the most general anisotropic Gaussian Schell-model (AGSM) beams, which constitute a ten-parameter family. Our approach is based on symmetry considerations. Concepts and techniques familiar from the context of quantum mechanics in the two-dimensional plane are used to exploit the Sp(4, R) dynamical symmetry underlying the AGSM problem. We take advantage of the fact that the symplectic group of first-order optical system acts unitarily through the metaplectic operators on the Hilbert space of wave amplitudes over the transverse plane, and, using the Iwasawa decomposition for the metaplectic operator and the classic theorem of Williamson on the normal forms of positive definite symmetric matrices under linear canonical transformations, we demonstrate the unitary equivalence of the AGSM problem to a separable problem earlier studied by Li and Wolf [Opt. Lett. 7, 256 (1982)] and Gori and Guattari [Opt. Commun. 48, 7 (1983)]. This conn ction enables one to write down, almost by inspection, the coherent-mode decomposition of the general AGSM beam. A universal feature of the eigenvalue spectrum of the AGSM family is noted.

Classical screw theory: A fresh look using dual eigenvalue approach

This paper presents a novel algebraic formulation of the central problem of screw theory, namely the determination of the principal screws of a given system. Using the algebra of dual numbers, it shows that the principal screws can be determined via the solution of a generalised eigenproblem of two real, symmetric matrices. This approach allows the study of the principal screws of the general screw systems associated with a manipulator of arbitrary geometry in terms of closed-form expressions of its architecture and configuration parameters. The formulation is illustrated with examples of practical manipulators.