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Immanant Of A Matrix
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent. Let \lambda=(\lambda_1,\lambda_2,\ldots) be a partition of an integer n and let \chi_\lambda be the corresponding irreducible representation-theoretic character of the symmetric group S_n. The ''immanant'' of an n\times n matrix A=(a_) associated with the character \chi_\lambda is defined as the expression :\operatorname_\lambda(A)=\sum_ \chi_\lambda(\sigma) a_ a_ \cdots a_ = \sum_ \chi_\lambda(\sigma) \prod_^ a_. Examples The determinant is a special case of the immanant, where \chi_\lambda is the alternating character \sgn, of ''S''''n'', defined by the parity of a permutation. The permanent is the case where \chi_\lambda is the trivial character, which is identically equal to 1. For example, for 3 \times 3 matrices, there are three irreducible representations of S_3, as shown in the character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Of A Permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of ''X'' is fixed, the parity (oddness or evenness) of a permutation \sigma of ''X'' can be defined as the parity of the number of inversions for ''σ'', i.e., of pairs of elements ''x'', ''y'' of ''X'' such that and . The sign, signature, or signum of a permutation ''σ'' is denoted sgn(''σ'') and defined as +1 if ''σ'' is even and −1 if ''σ'' is odd. The signature defines the alternating character of the symmetric group S''n''. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (''ε''''σ''), which is defined for all maps from ''X'' to ''X'', and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as : where ''N''('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Transactions Of The Royal Society A
''Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'' is a fortnightly peer-reviewed scientific journal published by the Royal Society. It publishes original research and review content in a wide range of physical scientific disciplines. Articles can be accessed online a few months prior to the printed journal. All articles become freely accessible two years after their publication date. The current editor-in-chief is John Dainton. Overview ''Philosophical Transactions of the Royal Society A'' publishes themed journal issues on topics of current scientific importance and general interest within the physical, mathematical and engineering sciences, edited by leading authorities and comprising original research, reviews and opinions from prominent researchers. Past issue titles include "Supercritical fluids - green solvents for green chemistry?", "Tsunamis: Bridging science, engineering and society", "Spatial transformations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gram Matrix
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\rangle., p.441, Theorem 7.2.10 If the vectors v_1,\dots, v_n are the columns of matrix X then the Gram matrix is X^\dagger X in the general case that the vector coordinates are complex numbers, which simplifies to X^\top X for the case that the vector coordinates are real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. Examples For finite-dimensional real vectors in \mathbb^n with the usual Euclidean dot product, the Gram matrix is G = V^\top V, where V is a matrix whose columns are the vectors v_k and V^\top is its transpose whose rows are the vectors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of The Symmetric Group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group S''n'' has order ''n''!. Its conjugacy classes are labeled by partitions of ''n''. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of ''n''. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of ''n'' or equivalently Young diagrams of size ''n''. Each such irreducible representation can in fact be realized over the integers (every permutation acting by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions a_ (x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Form
In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately K- linear in each of its k arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces. A multilinear k-form on V over \R is called a (covariant) \boldsymbol-tensor, and the vector space of such forms is usually denoted \mathcal^k(V) or \mathcal^k(V). Tensor product Given a k-tensor f\in\mathcal^k(V) and an \ell-tensor g\in\mathcal^\ell(V), a product f\otimes g\in\mathcal^(V), known as the tensor product, can be defined by the property : (f\otimes g)(v_1,\ldots,v_k,v_,\ldots, v_)=f(v_1,\ldots,v_k)g(v_,\ldots, v_), for all v_1,\ldots,v_\in V. The tensor product of multilinear forms is not commutative; however it is bilinear and associative: : f\otimes(ag_1+bg_2)=a(f\otimes g_1)+b( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Character
In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ( Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of ''V'' to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Character
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of ''X'' is fixed, the parity (oddness or evenness) of a permutation \sigma of ''X'' can be defined as the parity of the number of inversions for ''σ'', i.e., of pairs of elements ''x'', ''y'' of ''X'' such that and . The sign, signature, or signum of a permutation ''σ'' is denoted sgn(''σ'') and defined as +1 if ''σ'' is even and −1 if ''σ'' is odd. The signature defines the alternating character of the symmetric group S''n''. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (''ε''''σ''), which is defined for all maps from ''X'' to ''X'', and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as : where ''N''('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dudley E
Dudley ( , ) is a market town in the West Midlands, England, southeast of Wolverhampton and northwest of Birmingham. Historically part of Worcestershire, the town is the administrative centre of the Metropolitan Borough of Dudley. In the 2011 census, it had a population of 79,379. The wider Metropolitan Borough had a population of 312,900. In 2014, the borough council adopted a slogan describing Dudley as the capital of the Black Country, a title by which it had long been informally known. Originally a market town, Dudley was one of the birthplaces of the Industrial Revolution and grew into an industrial centre in the 19th century with its iron, coal, and limestone industries before their decline and the relocation of its commercial centre to the nearby Merry Hill Shopping Centre in the 1980s. Tourist attractions include Dudley Zoo and Castle, the 12th century priory ruins, and the Black Country Living Museum. History Early history Dudley has a history dating back to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
