Orthogonal matrix | EPFL Graph Search

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm{T} Q = Q Q^\mathrm{T} = I, where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: Q^\mathrm{T}=Q^{-1}, where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotatio

Application aux équations aux dérivées partielles d'une méthode par point fixe et le problème des deux puits

Sébastien Basterrechea

This thesis consists of two parts. The first part is about a variant of Banach's fixed point theorem and its applications to several partial differential equations (PDE's), abstractly of the form [ \mathcal Lu + \mathcal Q(u) = f.] The main result of this first part asserts that an equation having this form admits a solution if the datum

f

satisfies a certain smallness assumption. This result (we call it \emph{the fixed point method}) is relatively simple to use and can be applied to a large variety of PDE's. The downside is that it guarantees the existence of solutions only for "small" data. The equations we deal with are Jacobian equations, non-linear elliptic PDE's, transport problems and the semi-linear wave equation. The second part of the thesis treats the

\emph{two well problem}

in two dimensions [ \nabla u \in \mathbb{S}_A \cup \mathbb{S}_B\quad \text{almost everywhere in}\ \Omega, ] [ u = u_0\quad \text{on}\ \partial\Omega. ] For the non-degenerate case

\det(A),\det(B) \neq 0

, we show a non-existence result for piecewise regular solutions if

A

and

B

are non-orthogonal. For the degenerate and semi-degenerate cases, we give a characterisation for the rank-one convex hull of

\mathbb{S}_A \cup \mathbb{S}_B

and several existence results for Lipschitz and piecewise affine solutions. Finally, for each case, we construct several explicit non-trivial solutions for well-chosen boundary conditions

u_0

EPFL2015

Differential geometry applied to crystallography

Philippe Kocian

The mathematical facet of modern crystallography is essentially based on analytical geometry, linear algebra as well as group theory. This study endeavours to approach the geometry and symmetry of crystals using the tools furnished by differential geometry and the theory of Lie groups. These two branches of mathematics being little known to crystallographers, the pertinent definitions such as differentiable manifold, tangent space or metric tensor or even isometries on a manifold together with some important results are given first. The example of euclidean space, taken as riemannian manifold, is treated, in order to show that the affine aspect of this space is not at all an axiom but the consequence of the euclidean nature of the manifold. Attention is then directed to a particular subgroup of the group of euclidean isometries, namely that of translations. This has the property of a Lie group and it turns out that the action of its elements, as well as those of its Lie algebra, plays an important role in generating a lattice on a manifold and in its tangent space, too. In particular, it is pointed out that one and only one finite and free module of the Lie algebra of the group of translations can generate both, modulated and non-modulated lattices. This last classification therefore appears continuous rather than black and white and is entirely determined by the parametrisation considered. Since a lattice in a tangent space has the properties of a vector space, it always possesses the structure of a finite, free module, which shows that the assignment of aperiodicity to modulated structures is quite subjective, even unmotivated. Thanks to the concept of representation of a lattice or a crystal in a tangent space, novel definitions of the notions of symmetry operation of a space group and point symmetry operation, as well as symmetry element and intrinsic translation arise; they altogether naturally blend into the framework of differential geometry. In order to conveniently pass from one representation of a crystal in one tangent space to another or to the structure on a manifold, an equivalence relation on the tangent bundle of the manifold is introduced. This relation furthermore allows to extend the concept of symmetry operation to the tangent bundle; this extension furnishes, particularly in the euclidean case, a very practical way of representing symmetry operations of space groups completely devoid of any dependence on an origin, or, in other words, in which each and every point may be considered the origin. The investigation of the group of translations having being completed, the study of the linear parts of the isometries comes naturally. Based on the fact that the set of linear parts possesses the structure of a Lie group, several results are proven in a rigorous manner, such as the fact that a rotation angle of π/3 is incompatible with a three-dimensional cubic lattice. Procedures for determining different crystal systems in function of the type of rotation are laid out by way of the study of orthogonal matrices and their relation to the matrix associated with the type of system. Finally, the description of a crystal by its diffraction patterns is taken on. It is shown that the general aspect of such a pattern is directly linked to the action of that free and finite module of the Lie algebra of translations which generates a lattice on a manifold. In the case of modulated crystals, it is demonstrated that the appearance of supplementary spots is caused by the geometry, i.e. by the parametrisation of the manifold in which the crystal exists and not by the action of the module in the Lie algebra. Thus, there exists a neat separation: the geometrical aspect on the one hand, and the action of the group on the other. As the last topic, other ways of interpreting the diffraction pattern of a modulated structure are laid out in order to argue that mere experimental data do not warrant the uniqueness of a model. The goal of this study is by no means an attempt at overthrowing existing structural models such as the superspace-formalism or at revolutionising the methods for determining structures, but is rather aimed at sustaining that the definition of certain notions becomes thoroughly natural within the appropriate mathematical framework, and, that the term aperiodicity assigned to modulated structures no longer has a true meaning.

EPFL2009

Bayesian adaptation of chaos representations using variational inference and sampling on geodesics

Panagiotis Tsilifis

A novel approach is presented for constructing polynomial chaos representations of scalar quantities of interest (QoI) that extends previously developed methods for adaptation in Homogeneous Chaos spaces. In this work, we develop a Bayesian formulation of the problem that characterizes the posterior distributions of the series coefficients and the adaptation rotation matrix acting on the Gaussian input variables. The adaptation matrix is thus construed as a new parameter of the map from input to QoI, estimated through Bayesian inference. For the computation of the coefficients' posterior distribution, we use a variational inference approach that approximates the posterior with a member of the same exponential family as the prior, such that it minimizes a Kuilback-Leibler criterion. On the other hand, the posterior distribution of the rotation matrix is explored by employing a Geodesic Monte Carlo sampling approach, consisting of a variation of the Hamiltonian Monte Carlo algorithm for embedded manifolds, in our case, the Stiefel manifold of orthonormal matrices. The performance of our method is demonstrated through a series of numerical examples, including the problem of multiphase flow in heterogeneous porous media.

ROYAL SOC2018