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Schur–Horn Theorem
In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem. Statement Theorem. Let \mathbf=\_^N and \mathbf=\_^N be two sequences of real numbers arranged in a non-increasing order. There is a Hermitian matrix with diagonal values \_^N and eigenvalues \_^N if and only if : \sum_^n d_i \leq \sum_^n \lambda_i \qquad n=1,2,\ldots,N and : \sum_^N d_i= \sum_^N \lambda_i. Polyhedral geometry perspective Permutation polytope generated by a vector The permutation polytope generated by \tilde = (x_1, x_2,\ldots, x_n) \in \mathbb^n denoted by \mathcal_ is defined as the convex hull of the set \. Here S_n denotes the symmetr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff–von Neumann Theorem
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1, Thus, a doubly stochastic matrix is both left stochastic and right stochastic. Indeed, any matrix that is both left and right stochastic must be square: if every row sums to one then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal. Birkhoff polytope The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal one. (There are 2n-1 constraints rather than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Linear Algebra
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual differenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terry Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to ethnic Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers. He is widely regarded as one of the greatest living mathematicians and has been referred to as the " Mozart of mathematics". Life and career Family Tao's parents are first-generation immigrants from Hong Kong to Australia.''Wen Wei Po'', Page A4, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horn, Alfred
Alfred Horn (February 17, 1918 – April 16, 2001) was an American mathematician notable for his work in lattice theory and universal algebra. His 1951 paper "On sentences which are true of direct unions of algebras" described Horn clauses and Horn sentences, which later would form the foundation of logic programming. Biography Horn was born on Lower East Side, Manhattan. His parents were both deaf, and his father died when Horn was three years old. At this point, the children moved in with their grandparents on the mother's side. They would later move to Brooklyn where Horn spent most of his childhood, raised by his extended family. Horn attended the City College of New York, and later, New York University where he earned a Master's degree in mathematics. He went on to earn his Ph.D. at University of California, Berkeley in 1946. A year later, he started work at the University of California, Los Angeles, where he stayed until his retirement in 1988. He died in 2001 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur, Issai
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the University of Bonn, professor in 1919. As a student of Ferdinand Georg Frobenius, he worked on group representations (the subject with which he is most closely associated), but also in combinatorics and number theory and even theoretical physics. He is perhaps best known today for his result on the existence of the Schur decomposition and for his work on group representations (Schur's lemma). Schur published under the name of both I. Schur, and J. Schur, the latter especially in ''Journal für die reine und angewandte Mathematik''. This has led to some confusion. Childhood Issai Schur was born into a Jewish family, the son of the businessman Moses Schur and his wife Golde Schur (née Landau). He was born in Mogilev on the Dnieper Riv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Map
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. Formal definition Let ''M'' be a manifold with symplectic form ω. Suppose that a Lie group ''G'' acts on ''M'' via symplectomorphisms (that is, the action of each ''g'' in ''G'' preserves ω). Let \mathfrak be the Lie algebra of ''G'', \mathfrak^* its dual, and :\langle, \rangle : \mathfrak^* \times \mathfrak \to \mathbf the pairing between the two. Any ξ in \mathfrak induces a vector field ρ(ξ) on ''M'' describing the infinitesimal action of ξ. To be precise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Subgroup
In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate. Examples * For a finite field ''F'', the group of diagonal matrices \begin a & 0 \\ 0 & b \end where ''a'' and ''b'' are elements of ''F*''. This is called the split Cartan subgroup of GL2(''F''). * For a finite field ''F'', every maximal commutative semisimple subgroup of GL2(''F'') is a Cartan subgroup (and conversely). See also *Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ... References * * * * {{algebra-stub Algebraic geometry Linear algebraic groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
U(n)-equivariance Of Isomorphism
U or u, is the twenty-first and sixth-to-last letter and fifth vowel letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''u'' (pronounced ), plural ''ues''. History U derives from the Semitic waw, as does F, and later, Y, W, and V. Its oldest ancestor goes to Egyptian hieroglyphics, and is probably from a hieroglyph of a mace or fowl, representing the sound v.html"_;"title="Voiced_labiodental_fricative.html"_;"title="nowiki/>Voiced_labiodental_fricative">v">Voiced_labiodental_fricative.html"_;"title="nowiki/>Voiced_labiodental_fricative">vor_the_sound_[Voiced_labial–velar_approximant.html" ;"title="Voiced_labiodental_fricative">v.html" ;"title="Voiced_labiodental_fricative.html" ;"title="nowiki/>Voiced labiodental fricative">v">Voiced_labiodental_fricative.html" ;"title="nowiki/>Voiced labiodental fricative">vor the sound [Voiced labial–velar approximant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coadjoint Action
In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If \mathfrak denotes the Lie algebra of G, the corresponding action of G on \mathfrak^*, the dual space to \mathfrak, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on G. The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups G a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of G are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of G, which again may be complicated, while the orbits are relatively tractable. Formal definition Let G be a Lie group and \mathfrak be its Lie algebra. Let \mathrm : G \rightarrow \mathrm(\mathfrak) denote the adjoint re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew-Hermitian
__NOTOC__ In linear algebra, a square matrix with Complex number, complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relation where A^\textsf denotes the conjugate transpose of the matrix A. In component form, this means that for all indices i and j, where a_ is the element in the j-th row and i-th column of A, and the overline denotes complex conjugate, complex conjugation. Skew-Hermitian matrices can be understood as the complex versions of real Skew-symmetric matrix, skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers., §4.1.2 The set of all skew-Hermitian n \times n matrices forms the u(n) Lie algebra, which corresponds to the Lie group Unitary group, U(n). The concept can be generalized to include linear transformations of any complex number, complex vector space with a sesquilinear Norm (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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