Jordan matrixIn the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form: Every Jordan block is specified by its dimension n and its eigenvalue , and is denoted as Jλ,n. It is an matrix of zeroes everywhere except for the diagonal, which is filled with and for the superdiagonal, which is composed of ones.
Canonical formIn mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.
UnipotentEn mathématiques, un élément unipotent r d'un anneau unitaire R est un tel que r − 1 est un élément nilpotent ; en d'autres termes, (r − 1)n vaut zéro pour n assez grand. En particulier, une matrice carrée M est une matrice unipotente si et seulement si son polynôme caractéristique P(t) est une puissance de t − 1. Ainsi, toutes les valeurs propres d'une matrice unipotente valent 1. Le terme quasi-unipotent signifie qu'une certaine puissance de l'élément est unipotente.
Central simple algebraIn ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A which is simple, and for which the center is exactly K. (Note that not every simple algebra is a central simple algebra over its center: for instance, if K is a field of characteristic 0, then the Weyl algebra is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.
Espace topologique irréductibleEn topologie, un espace irréductible est un espace topologique non vide qui ne peut pas se décomposer en (c'est-à-dire s'écrire comme réunion de) deux parties fermées strictement plus petites. Ce type d'espaces apparaît (et est utilisé) surtout en géométrie algébrique, où l'irréductibilité est une des propriétés topologiques basiques.
Ultraconnected spaceIn mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T1 space with more than one point is ultraconnected. Every ultraconnected space is path-connected (but not necessarily arc connected). If and are two points of and is a point in the intersection , the function defined by if , and if , is a continuous path between and .
Drapeau (mathématiques)En mathématiques, un drapeau d'un espace vectoriel E de dimension finie est une suite finie strictement croissante de sous-espaces vectoriels de E, commençant par l'espace nul {0} et se terminant par l'espace total E : Si n est la dimension de E, les dimensions successives des sous-espaces E forment une suite finie strictement croissante d'entiers naturels : Si d = i pour tout i (donc entre autres si k = n), alors le drapeau est dit total ou complet. À toute base (e, ...
Suite de compositionLa notion de suite de composition est une notion de théorie des groupes. Elle permet, dans un sens qui sera précisé, de considérer un groupe comme « composé » de certains de ses sous-groupes. Soient G un groupe et e son élément neutre. On appelle suite de composition de G toute suite finie (G_0, G_1, ..., G_r) de sous-groupes de G telle queet que, pour tout i ∈ {0, 1, ..., r – 1}, G_i+1 soit sous-groupe normal de G_i.Les quotients G_i/G_i+1 sont appelés les quotients de la suite. Soient Σ_1 = (G_0, G_1, ...
Cartan subalgebraIn mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., ), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism is semisimple (i.
