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Birkhoff Factorization
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by , is a generalization of the LU decomposition (i.e. Gauss elimination) to loop groups. The factorization of an invertible matrix M\in\mathrm_n(\mathbb ,z^ with coefficients that are Laurent polynomials in z is given by a product M=M^M^M^, where M^ has entries that are polynomials in z, M^=\mathrm(z^, z^,...,z^) is diagonal with k_i\in\mathbb for 1\leq i\leq n and k_1\geq k_2\geq ...\geq k_n, and M^ has entries that are polynomials in z^. For a generic matrix we have M^=\mathrm. Birkhoff factorization implies the Birkhoff–Grothendieck theorem of that vector bundles over the projective line are sums of line bundles. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to . Birkhoff factorization follows from the Bruhat decomposition for affine Kac–Moody groups (or loop groups), and conversely the Bruhat decomposition for the affine g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LU Decomposition
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. It is also sometimes referred to as LR decomposition (factors into left and right triangular matrices). The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938, who first wrote product equation LU=A=h^Tg (The last form in his alternate yet equivalent matrix notation appears as g\times h. ) Definitions Let ''A'' be a square matrix. An LU factorization refers to expression of ''A'' into product of two facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their inverse. Definition An -by- square matrix is called invertible if there exists an -by- square matrix such that\mathbf = \mathbf = \mathbf_n ,where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. Over a field, a square matrix that is ''not'' invertible is called singular or deg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Polynomials
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in X form a ring denoted \mathbb , X^/math>. They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables. Laurent polynomials are of particular importance in the study of complex variables. Definition A Laurent polynomial with coefficients in a field \mathbb is an expression of the form : p = \sum_k p_k X^k, \quad p_k \in \mathbb where X is a formal variable, the summation index k is an integer (not necessarily positive) and only finitely many coefficients p_ are non-zero. Two Laurent polynomials are equal if their coefficients are equal. Such expressions can be added, multiplied, and brought back to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Group
In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over ''K''). Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, finitely generated infinite torsion groups). Definition and basic examples A group ''G'' is said to be ''linear'' if there exists a field ''K'', an integer ''d'' and an injective homomorphism from ''G'' to the general linear group GL''d''(''K'') (a faithful linear representation of dimension ''d'' over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruhat Decomposition
In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) G=BWB of certain algebraic groups G=BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this. More generally, any group with a (''B'', ''N'') pair has a Bruhat decomposition. Definitions *G is a connected, reductive algebraic group over an algebraically closed field. *B is a Borel subgroup of G *W is a Weyl group of G corresponding to a maximal torus of B. The Bruhat decomposition of G is the decomposition :G=BWB =\bigsqcup_BwB of G as a disjoint union of double cosets of B parameterized by the elements of the Weyl group W. (Note that although W is not in general a subgroup of G, the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Group
In mathematics, a loop group (not to be confused with a loop) is a group of loops in a topological group ''G'' with multiplication defined pointwise. Definition In its most general form a loop group is a group of continuous mappings from a manifold to a topological group . More specifically, let , the circle in the complex plane, and let denote the space of continuous maps , i.e. :LG = \, equipped with the compact-open topology. An element of is called a ''loop'' in . Pointwise multiplication of such loops gives the structure of a topological group. Parametrize with , :\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G, and define multiplication in by :(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta). Associativity follows from associativity in . The inverse is given by :\gamma^:\gamma^(\theta) \equiv \gamma(\theta)^, and the identity by :e:\theta \mapsto e \in G. The space is called the free loop group on . A loop group is any subgroup of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff Decomposition (other) .
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Birkhoff decomposition refers to two different mathematical concepts: * The Birkhoff factorization, introduced by George David Birkhoff at 1909, is the presentation of an invertible matrix with polynomial coefficients as a product of three matrices. * The Birkhoff - von Neumann decomposition, introduced by Garrett Birkhoff (George's son) at 1946, is the presentation of a bistochastic matrix as a convex sum of permutation matrices. It can be found by the Birkhoff algorithm Birkhoff's algorithm (also called Birkhoff-von-Neumann algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices. It was published by Garrett Birkhoff in 1946. It has many applications. One s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Hilbert Problem
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others. The Riemann problem Suppose that \Sigma is a smooth, simple, closed contour in the complex plane. Divide the plane into two parts denoted by \Sigma_ (the inside) and \Sigma_ (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation, was that of finding a function :M_+(t) = u(t) + i v(t), analytic inside \Sigma_, such that the boundary values of M_+ along \Sigma satisfy the equation :a(t)u(t) - b(t)v(t) = c(t), for t \in \Sigma, where a(t), b(t) and c(t) are given real-valued functions. For example, in the special case where a = 1, b=0 and \Sigma is a circle, the problem reduces to deriving ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * '' Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
