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Thesis (Ph.D.)--Auburn University, 2005. / Abstract. Vita. Includes bibliographic references (ℓ. )

Thesis (Ph. D.)--University of Washington, 1994. / Vita. Includes bibliographical references (leaves [139]-141).

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The variety of principal minors of nxn symmetric matrices, denoted Zn, can be described naturally as a projection from the Lagrangian Grassmannian. Moreover, Zn is invariant under the action of a group G C GL(2n) isomorphic to (SL(2)xn) x Sn. One may use this symmetry to study the defining ideal of Zn as a G-module via a coupling of classical representation theory and geometry. The need for the equations in the defining ideal comes from applications in matrix theory, probability theory, spectral graph theory and statistical physics. I describe an irreducible G-module of degree 4 polynomials called the hyperdeterminantal module (which is constructed as the span of the G-orbit of Cayley's hyperdeterminant of format 2 x 2 x 2) and show that it that cuts out Zn set theoretically. This result solves the set-theoretic version of a conjecture of Holtz and Sturmfels and gives a collection of necessary and sufficient conditions for when it is possible for a given vector of length 2n to be the principal minors of a symmetric n x n matrix. In addition to solving the Holtz and Sturmfels conjecture, I study Zn as a prototypical G-variety. As a result, I exhibit the use of and further develop techniques from classical representation theory and geometry for studying G-varieties.

Cette thèse traite de l'étude de méthodes de diagonalisation conjointe de matrices complexes, en vue de la séparation de sources, que ce soit dans le domaine des télécommunications numériques ou de la radioastronomie. Après avoir présenté les motivations qui ont poussé cette étude, nous faisons un bref état de l'art dans le domaine. Le problème de la diagonalisation conjointe, ainsi que celui de la séparation de source sont rappelés, et un lien entre ces deux sujets est établi. Par la suite, plusieurs algorithmes itératifs sont développés. Dans un premier temps, des méthodes utilisant une mise à jour de la matrice de séparation, de type gradient, sont présentées. Elles sont basées sur des approximations judicieuses du critère considéré. Afin d'améliorer la vitesse de convergence, une méthode utilisant un calcul du pas optimal est présentée, et plusieurs variantes de ce calcul, basées sur les approximations faites précédemment, sont développées. Deux autres approches sont ensuite introduites. La première détermine la matrice de séparation de manière analytique, en calculant algébriquement les termes composant la matrice de mise à jour par paire à partir d'un système d'équations linéaire. La deuxième estime récursivement la matrice de mélange, en se basant sur une méthode de moindres carrés alternés. Afin d'améliorer la vitesse de convergence, une recherche de pas d'adaptation linéaire est proposée. Ces méthodes sont alors validées sur un problème de diagonalisation conjointe classique. Puis les algorithmes sont appliqués à la séparation de sources de signaux de télécommunication numérique, en utilisant des statistiques d'ordre deux ou supérieur. Des comparaisons sont également effectuées avec des méthodes standards. La deuxième application concerne l'élimination des interférences terrestres à partir de l'estimation de l'espace associé, afin d'observer au mieux des sources cosmiques, issues de données de station LOFAR. / This thesis deals with the study of joint diagonalization of complex matrices methods for source separation, wether in the field of numerical telecommunications and radioastronomy. After having introduced the motivations that drove this study, we present a brief state-of-the-art in the field. The joint diagonalization and source separation problems are reminded, and a link between these two themes is established. Thereafter, several iterative algorithms are developed. First, methods using a gradient-like update of the separation matrix are introduced. They are based on wise approximations of the considered criterion. In order to improve the convergence speed, a method using a computation of an optimal step size is presented, and variations around this computation, based on the previously introduced approximations are done. Two other approaches are then introduced. The first one analytically determines the separation matrix, by algebraically computing the terms composing the update matrix pairwise from a linear equation system. The second one recursively estimates the mixing matrix, based on an alternating least squares method. In order to enhance the convergence speed, a seek of an enhanced line search algorithm is proposed. These methods are then validated on a classical joint diagonalization problem. Aterwards, these algorithms are applied to the source separation of numerical communication signals, while using second or higher order statistics. Comparisons are also made with well-known methods. The second application relates to elimination of rterrestrial interferences from the estimation of the associated space in order to observe at best cosmic sources from LOFAR station data.

Involutions and fixed-point-free involutions arise naturally as representatives for certain Borel orbits in invertible matrices. Similarly, partial involutions and partial fixed-point-free involutions represent certain Borel orbits in matrices which are not necessarily invertible. Inclusion relations among Borel orbit closures induce a partial order on these discrete parameterizing sets. In this dissertation we investigate the associated order complex of these posets. In particular, we prove that the order complex of the Bruhat poset of Borel orbit closures is shellable in symmetric as well as skew-symmetric matrices. / acase@tulane.edu

Let G be a simple graph on n vertices, and let S(G) be the class of all real-valued symmetric nxn matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The smallest rank achieved by a matrix in S(G) is called the minimum rank of G, denoted mr(G). The maximum nullity achieved by a matrix in S(G) is denoted M(G). For each graph G, there is an associated minimum rank class, MR(G) consisting of all matrices A in S(G) with rank A = mr(G). Although no restrictions are applied to the diagonal entries of matrices in S(G), sometimes diagonal entries corresponding to specific vertices of G must be zero for all matrices in MR(G). These vertices are known as nil vertices (see [6]). In this paper I discuss some basic results about nil vertices in general and nil vertices in cographs and prove that cographs with a nil vertex of a particular form contain two other nil vertices symmetric to the first. I discuss several open questions relating to these results and a counterexample. I prove that for all cographs G without an induced complete tripartite graph with independent sets all of size 3, the zero-forcing number Z(G), a graph theoretic parameter, is equal to M(G). In fact this result holds for a slightly larger class of cographs and in particular holds for all threshold graphs. Lastly, I prove that the maximum of the minimum ranks of all cographs on n vertices is the floor of 2n/3.

El tema principal de esta tesis es el desarrollo de técnicas de actualización de precondicionadores para resolver sistemas lineales de gran tamaño y dispersos Ax=b mediante el uso de métodos iterativos de Krylov. Se consideran dos tipos interesantes de problemas. En el primero se estudia la solución iterativa de sistemas lineales no singulares y antisimétricos, donde la matriz de coeficientes A tiene parte antisimétrica de rango bajo o puede aproximarse bien con una matriz antisimétrica de rango bajo. Sistemas como este surgen de la discretización de PDEs con ciertas condiciones de frontera de Neumann, la discretización de ecuaciones integrales y métodos de puntos interiores, por ejemplo, el problema de Bratu y la ecuación integral de Love. El segundo tipo de sistemas lineales considerados son problemas de mínimos cuadrados (LS) que se resuelven considerando la solución del sistema equivalente de ecuaciones normales. Concretamente, consideramos la solución de problemas LS modificados y de rango incompleto. Por problema LS modificado se entiende que el conjunto de ecuaciones lineales se actualiza con alguna información nueva, se agrega una nueva variable o, por el contrario, se elimina alguna información o variable del conjunto. En los problemas LS de rango deficiente, la matriz de coeficientes no tiene rango completo, lo que dificulta el cálculo de una factorización incompleta de las ecuaciones normales. Los problemas LS surgen en muchas aplicaciones a gran escala de la ciencia y la ingeniería como, por ejemplo, redes neuronales, programación lineal, sismología de exploración o procesamiento de imágenes. Los precondicionadores directos para métodos iterativos usados habitualmente son las factorizaciones incompletas LU, o de Cholesky cuando la matriz es simétrica definida positiva. La principal contribución de esta tesis es el desarrollo de técnicas de actualización de precondicionadores. Básicamente, el método consiste en el cálculo de una descomposición incompleta para un sistema lineal aumentado equivalente, que se utiliza como precondicionador para el problema original. El estudio teórico y los resultados numéricos presentados en esta tesis muestran el rendimiento de la técnica de precondicionamiento propuesta y su competitividad en comparación con otros métodos disponibles en la literatura para calcular precondicionadores para los problemas estudiados. / The main topic of this thesis is updating preconditioners for solving large sparse linear systems Ax=b by using Krylov iterative methods. Two interesting types of problems are considered. In the first one is studied the iterative solution of non-singular, non-symmetric linear systems where the coefficient matrix A has a skew-symmetric part of low-rank or can be well approximated with a skew-symmetric low-rank matrix. Systems like this arise from the discretization of PDEs with certain Neumann boundary conditions, the discretization of integral equations as well as path following methods, for example, the Bratu problem and the Love's integral equation. The second type of linear systems considered are least squares (LS) problems that are solved by considering the solution of the equivalent normal equations system. More precisely, we consider the solution of modified and rank deficient LS problems. By modified LS problem, it is understood that the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Rank deficient LS problems are characterized by a coefficient matrix that has not full rank, which makes difficult the computation of an incomplete factorization of the normal equations. LS problems arise in many large-scale applications of the science and engineering as for instance neural networks, linear programming, exploration seismology or image processing. Usually, incomplete LU or incomplete Cholesky factorization are used as preconditioners for iterative methods. The main contribution of this thesis is the development of a technique for updating preconditioners by bordering. It consists in the computation of an approximate decomposition for an equivalent augmented linear system, that is used as preconditioner for the original problem. The theoretical study and the results of the numerical experiments presented in this thesis show the performance of the preconditioner technique proposed and its competitiveness compared with other methods available in the literature for computing preconditioners for the problems studied. / El tema principal d'esta tesi és actualitzar precondicionadors per a resoldre sistemes lineals grans i buits Ax=b per mitjà de l'ús de mètodes iteratius de Krylov. Es consideren dos tipus interessants de problemes. En el primer s'estudia la solució iterativa de sistemes lineals no singulars i antisimètrics, on la matriu de coeficients A té una part antisimètrica de baix rang, o bé pot aproximar-se amb una matriu antisimètrica de baix rang. Sistemes com este sorgixen de la discretització de PDEs amb certes condicions de frontera de Neumann, la discretització d'equacions integrals i mètodes de punts interiors, per exemple, el problema de Bratu i l'equació integral de Love. El segon tipus de sistemes lineals considerats, són problemes de mínims quadrats (LS) que es resolen considerant la solució del sistema equivalent d'equacions normals. Concretament, considerem la solució de problemes de LS modificats i de rang incomplet. Per problema LS modificat, s'entén que el conjunt d'equacions lineals s'actualitza amb alguna informació nova, s'agrega una nova variable o, al contrari, s'elimina alguna informació o variable del conjunt. En els problemes LS de rang deficient, la matriu de coeficients no té rang complet, la qual cosa dificultata el calcul d'una factorització incompleta de les equacions normals. Els problemes LS sorgixen en moltes aplicacions a gran escala de la ciència i l'enginyeria com, per exemple, xarxes neuronals, programació lineal, sismologia d'exploració o processament d'imatges. Els precondicionadors directes per a mètodes iteratius utilitzats més a sovint són les factoritzacions incompletes tipus ILU, o la factorització incompleta de Cholesky quan la matriu és simètrica definida positiva. La principal contribució d'esta tesi és el desenvolupament de tècniques d'actualització de precondicionadors. Bàsicament, el mètode consistix en el càlcul d'una descomposició incompleta per a un sistema lineal augmentat equivalent, que s'utilitza com a precondicionador pel problema original. L'estudi teòric i els resultats numèrics presentats en esta tesi mostren el rendiment de la tècnica de precondicionament proposta i la seua competitivitat en comparació amb altres mètodes disponibles en la literatura per a calcular precondicionadors per als problemes considerats. / Guerrero Flores, DJ. (2018). On Updating Preconditioners for the Iterative Solution of Linear Systems [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/104923 / TESIS