yes / An expansion of row Markov matrices in terms of matrices related to permutations with repetitions, is introduced. It generalises the Birkhoff-von Neumann expansion of doubly stochastic matrices in terms of permutation matrices (without repetitions). An interpretation of the formalism in terms of sequences of integers that open random safes described by the Markov matrices, is presented. Various quantities that describe probabilities and correlations in this context, are discussed. The Gini index is used to quantify the sparsity (certainty) of various probability vectors. The formalism is used in the context of multipartite quantum systems with finite dimensional Hilbert space, which can be viewed as quantum permutations with repetitions or as quantum safes. The scalar product of row Markov matrices, the various Gini indices, etc, are novel probabilistic quantities that describe the statistics of multipartite quantum systems. Local and global Fourier transforms are used to de ne locally dual and also globally dual statistical quantities. The latter depend on off-diagonal elements that entangle (in general) the various components of the system. Examples which demonstrate these ideas are also presented.

Three different definitions for the determinant of a matrix over arbitrary lattices have been developed to determine which properties and relations were reminiscent of the determinant or permanent of elementary algebra. In each determinant there are properties concerning: the elements of the matrix in the expansion of its determinant; the determinant of a matrix and its transpose; a principle of duality for rows and columns; the interchange of rows and columns; the determinant of a matrix formed from another by a row or column meet of certain elements; and evaluations of certain special matrices. An expansion by row or column is given for one determinant and a lemma on inverses is proven in light of another. A preliminary section on Lattice Theory is also included. / Master of Science

Yes / Financial performance of insurance companies is captured by changes in rating grades. An insurer is susceptible to a rating transition which is a signal depicting current financial conditions. We employ Rating Transition Matrices (RTM) to analyse these transitions. Within this context, credit quality can either improve, remain stable or deteriorate as reflected by a rating upgrade or downgrade. We investigate rating trends and forecast rating transitions for UK insurers. We also provide insights into the effects of the global financial crisis on financial performance of UK insurance companies, as reflected by rating changes. Our analysis shows a significant degree of rating changes, as reflected by rating fluctuations in rating matrices. We conclude that insurers with higher (better) rating grades depict rating stability over the long-run. An unexpected but interested finding shows that insurers with good rating grades are nevertheless susceptible to rating fluctuations. General insurers are more likely to be rated and they demonstrate higher levels of rating grade variations over the period studied. Using comparative rating transition matrices, we find more variations in rating movements in the post-financial crisis period. We also conclude that general insurers reflect less stable rating outlooks compared to life and general insurers.

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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.

Dans cette thèse, on aborde trois thèmes : problème de sélection de colonnes dans une matrice, distance de Banach-Mazur au cube et estimation de la covariance de matrices aléatoires. Bien que les trois thèmes paraissent éloignés, les techniques utilisées se ressemblent tout au long de la thèse. Dans un premier lieu, nous généralisons le principe d'invertibilité restreinte de Bourgain-Tzafriri. Ce résultat permet d'extraire un "grand" bloc de colonnes linéairement indépendantes dans une matrice et d'estimer la plus petite valeur singulière de la matrice extraite. Nous proposons ensuite un algorithme déterministe pour extraire d'une matrice un bloc presque isométrique c’est à dire une sous-matrice dont les valeurs singulières sont proches de 1. Ce résultat nous permet de retrouver le meilleur résultat connu sur la célèbre conjecture de Kadison-Singer. Des applications à la théorie locale des espaces de Banach ainsi qu'à l'analyse harmonique sont déduites. Nous donnons une estimation de la distance de Banach-Mazur d'un corps convexe de Rn au cube de dimension n. Nous proposons une démarche plus élémentaire, basée sur le principe d'invertibilité restreinte, pour améliorer et simplifier les résultats précédents concernant ce problème. Plusieurs travaux ont été consacrés pour approcher la matrice de covariance d'un vecteur aléatoire par la matrice de covariance empirique. Nous étendons ce problème à un cadre matriciel et on répond à la question. Notre résultat peut être interprété comme une quantification de la loi des grands nombres pour des matrices aléatoires symétriques semi-définies positives. L'estimation obtenue s'applique à une large classe de matrices aléatoires / In this thesis, we address three themes : columns subset selection in a matrix, the Banach-Mazur distance to the cube and the estimation of the covariance of random matrices. Although the three themes seem distant, the techniques used are similar throughout the thesis. In the first place, we generalize the restricted invertibility principle of Bougain-Tzafriri. This result allows us to extract a "large" block of linearly independent columns inside a matrix and estimate the smallest singular value of the restricted matrix. We also propose a deterministic algorithm in order to extract an almost isometric block inside a matrix i.e a submatrix whose singular values are close to 1. This result allows us to recover the best known result on the Kadison-Singer conjecture. Applications to the local theory of Banach spaces as well as to harmonic analysis are deduced. We give an estimate of the Banach-Mazur distance between a symmetric convex body in Rn and the cube of dimension n. We propose an elementary approach, based on the restricted invertibility principle, in order to improve and simplify the previous results dealing with this problem. Several studies have been devoted to approximate the covariance matrix of a random vector by its sample covariance matrix. We extend this problem to a matrix setting and we answer the question. Our result can be interpreted as a quantified law of large numbers for positive semidefinite random matrices. The estimate we obtain, applies to a large class of random matrices

Este trabalho tem como objetivo central a elaboração, implementação e reflexão sobre uma sequência didática para o estudo de matrizes a partir de tranformações geométricas. A sequência didática pretende propiciar ao aluno um estudo que justifique as definições das operações entre matrizes e suas respectivas propriedades, a partir da observação e análise de algumas transformações geométricas, de modo a se refazer o processo histórico da definição e obtenção desses conceitos. Além disso apresenta algumas atividades de aplicação de matrizes, onde a composição e iteração de transformações geométricas no software Shapari geram algumas figuras fractais. Como metodologia de trabalho adotamos a engenharia didática para a elaboração, implementação e avaliação da didática proposta. O texto também apresenta a análise das referências sobre o ensino de matrizes e tranformações geométricas. Começando pelas orientações dos documentos oficiais e passando pela apresentação de diversos estudos sobre o tema delimitamos e justificamos a nossa proposta de ensino. Além disso, apresentamos um extrato sobre o conhecimento matemático envolvido no tema, de modo que sirva de base para o docente que implementar a seuência didática em sala de aula. / This work has as its main goal the formulation, implementation and contemplation of a didactie sequence to the study of matrices from geometric transformations. The didactic sequence intends to propitiate the student a study which justifies the definitions of operations between matrices and their respective properties, from the observation and analysis of some geometrie transformations, so that they are able to redo the historical process of the definition and the acquisition of these concepts. Besides that, it shows some activities to the application of matrices, in which the composition and interation of geometric transformations in the Shapari software generate some fractals. We have adopted Didactic Engineering as a methodology to the elaboration, implementation and evaluation of the proposed didactic sequence. The text also shows an analysis of the references about the teaching of matrices and the geometric transformations. We delimited and justified our teaching proposal by staring with orientations of official documents and showing a series of studies about the theme. Besides that, we show an excerpt about the mathematical knowledge involved in the theme, so that, it can be used as basis to the teacher who decides to implement the didactic sequence in the classroom.

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We consider three problems in this dissertation, all under the unifying theme of random matrix products. The first and second problems are concerned with weak convergence in stochastic matrices and circulant matrices, respectively, and the third is concerned with the numerical calculation of the Lyapunov exponent associated with some random Fibonacci sequences. Stochastic matrices are nonnegative matrices whose row sums are all equal to 1. They are most commonly encountered as transition matrices of Markov chains. Circulant matrices, on the other hand, are matrices where each row after the first is just the previous row cyclically shifted to the right by one position. Like stochastic matrices, circulant matrices are ubiquitous in the literature.In the first problem, we study the weak convergence of the convolution sequence mu to the n, where mu is a probability measure with support S sub mu inside the space S of d by d stochastic matrices, d greater than or equal to 3. Note that mu to the n is precisely the distribution of the product X sub 1 times X sub 2 times and so on times X sub n of the mu distributed independent random variables X sub 1, X sub 2, and so on, X sub n taking values in S. In [CR] Santanu Chakraborty and B.V. Rao introduced a cyclicity condition on S sub mu and showed that this condition is necessary and sufficient for mu to the n to not converge weakly when d is equal to 3 and the minimal rank r of the matrices in the closed semigroup S generated by S sub mu is 2. Here, we extend this result to any d bigger than 3. Moreover, we show that when the minimal rank r is not 2, this result does not always hold.The second problem is an investigation of weak convergence in another direction, namely the case when the probability measure mu's support S sub mu consists of d by d circulant matrices, d greater than or equal to 3, which are not necessarily nonnegative. The resulting semigroup S generated by S sub mu now lacking the nice property of compactness in the case of stochastic matrices, we assume tightness of the sequence mu to the n to analyze the problem. Our approach is based on the work of Mukherjea and his collaborators, who in [LM] and [DM] presented a method based on a bookkeeping of the possible structure of the compact kernel K of S.The third problem considered in this dissertation is the numerical determination of Lyapunov exponents of some random Fibonacci sequences, which are stochastic versions of the classical Fibonacci sequence f sub (n plus 1) equals f sub n plus f sub (n minus 1), n greater than or equal to 1, and f sub 0 equal f sub 1 equals 1, obtained by randomizing one or both signs on the right side of the defining equation and or adding a "growth parameter." These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent. Following techniques presented by Embree and Trefethen in their numerical paper [ET], we study the behavior of the Lyapunov exponents as a function of the probability p of choosing plus in the sign randomization.

Having origins in the increasingly popular Matrix Theory, the square root function of a matrix has received notable attention in recent years. In this thesis, we discuss some of the more common matrix functions and their general properties, but we specifically explore the square root function of a matrix and the most efficient method (Schur decomposition) of computing it. Calculating the square root of a 2×2 matrix by the Cayley-Hamilton Theorem is highlighted, along with square roots of positive semidefinite matrices and general square roots using the Jordan Canonical Form.