|
Coherent Algebra
A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix I and the all-ones matrix J. Definitions A subspace \mathcal of \mathrm_(\mathbb) is said to be a coherent algebra of order n if: * I, J \in \mathcal. * M^ \in \mathcal for all M \in \mathcal. * MN \in \mathcal and M \circ N \in \mathcal for all M, N \in \mathcal. A coherent algebra \mathcal is said to be: * ''Homogeneous'' if every matrix in \mathcal has a constant diagonal. * ''Commutative'' if \mathcal is commutative with respect to ordinary matrix multiplication. * ''Symmetric'' if every matrix in \mathcal is symmetric. The set \Gamma(\mathcal) of ''Schur-primitive matrices'' in a coherent algebra \mathcal is defined as \Gamma(\mathcal) := \ . Dually, the set \Lambda(\mathcal) of ''primitive matrices'' in a coherent algebra \mathcal is defined as \Lambda(\mathcal) := \ . Examples * The c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation. Finite groups For a finite group ''G'', the left regular representation λ (over a field ''K'') is a linear representation on the ''K''-vector space ''V'' freely generated by the elements of ''G'', i. e. they can be identified with a basis of ''V''. Given ''g'' ∈ ''G'', λ''g'' is the linear map determined by its action on the basis by left translation by ''g'', i.e. :\lambda_:h\mapsto gh,\texth\in G. For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given ''g'' ∈ ''G'', ρ''g'' is the linear map on ''V'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Association Scheme
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups. Definition An ''n''-class association scheme consists of a set ''X'' together with a partition ''S'' of ''X'' × ''X'' into ''n'' + 1 binary relations, ''R''0, ''R''1, ..., ''R''''n'' which satisfy: *R_ = \ and is called the identity relation. *Defining R^* := \, if ''R'' in ''S'', then ''R*'' in ''S'' *If (x,y) \in R_, the number of z \in X such that (x,z) \in R_ and (z,y) \in R_ is a constant p^k_ depending on i, j, k but not on the particular choice of x and y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal Ring
In mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when ''R'' is a commutative ring, ''R'' can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of ''R'' are principal, then ''R'' is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains. A commutative principal ideal ring which is also an integral domain is said to be a ''principal ideal domain'' (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain. General properties If ''R'' is a principal right ideal ring, then it is certainly a right Noetherian rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Association Scheme
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups. Definition An ''n''-class association scheme consists of a set ''X'' together with a partition ''S'' of ''X'' × ''X'' into ''n'' + 1 binary relations, ''R''0, ''R''1, ..., ''R''''n'' which satisfy: *R_ = \ and is called the identity relation. *Defining R^* := \, if ''R'' in ''S'', then ''R*'' in ''S'' *If (x,y) \in R_, the number of z \in X such that (x,z) \in R_ and (z,y) \in R_ is a constant p^k_ depending on i, j, k but not on the particular choice of x and y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bose–Mesner Algebra
In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative algebra. Among these rules are: :*the result of a product is also within the set of matrices, :*there is an identity matrix in the set, and :*taking products is commutative. Bose–Mesner algebras have applications in physics to spin models, and in statistics to the design of experiments. They are named for R. C. Bose and Dale Marsh Mesner. Definition Let ''X'' be a set of ''v'' elements. Consider a partition of the 2-element subsets of ''X'' into ''n'' non-empty subsets, ''R''1, ..., ''R''''n'' such that: * given an x \in X, the number of y \in X such that \ \in R_i depends only on i (and not on ''x''). This number will be denoted by vi, and * given x,y \in X wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product. The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the ''Zehfuss matrix'', and the ''Zehfuss product'', after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used. Definition If A is an matrix and B is a matrix, then the Kr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (set Theory)
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The number 9 is in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope '' acts transitively'' on its vertices, or that the vertices lie within a single '' symmetry orbit''. All vertices of a finite -dimensional isogonal figure exist on an -sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory. The pseudorhombicuboctahedronwhich is ''not'' isogonaldemonstrates that simply asserting that "all vertices look the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices and is denoted as . Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra. Notation This article will use the following notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph is a permutation of the vertex set , such that the pair of vertices form an edge if and only if the pair also form an edge. That is, it is a graph isomorphism from to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. Computational complexity Constructing the automorphism group is at least as difficult (in terms of its computational complexity) as solving the graph is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
