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Frobenius Theorem (real Division Algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: * (the real numbers) * (the complex numbers) * (the quaternions). These algebras have real dimension , and , respectively. Of these three algebras, and are commutative, but is not. Proof The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra. Introducing some notation * Let be the division algebra in question. * Let be the dimension of . * We identify the real multiples of with . * When we write for an element of , we tacitly assume that is contained in . * We can consider as a finite-dimensional - vector space. Any element of defines an endomorphism of by left-multiplication, we identify with that ... [...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] |
Minimal Polynomial (linear Algebra)
In linear algebra, the minimal polynomial of an matrix over a field is the monic polynomial over of least degree such that . Any other polynomial with is a (polynomial) multiple of . The following three statements are equivalent: # is a root of , # is a root of the characteristic polynomial of , # is an eigenvalue of matrix . The multiplicity of a root of is the largest power such that ''strictly'' contains . In other words, increasing the exponent up to will give ever larger kernels, but further increasing the exponent beyond will just give the same kernel. If the field is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in ) alone, in other words they may have irreducible polynomial factors of degree greater than . For irreducible polynomials one has similar equivalences: # divides , # divides , # the kernel of has dimension at least . # the kernel of has dimension at least . Like ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is '' locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octonions
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the sedenions. History The octonions were discovered in 1843 by John T. Graves, inspired by his fri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Division Algebra
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in . Subsequent proofs of the restrictions on the dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz's Theorem (normed Division Algebras)
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Hurwitz's theorem can refer to several theorems named after Adolf Hurwitz: * Hurwitz's theorem (complex analysis) * Riemann–Hurwitz formula in algebraic geometry * Hurwitz's theorem (composition algebras) on quadratic forms and nonassociative algebras * Hurwitz's automorphisms theorem on Riemann surfaces * Hurwitz's theorem (number theory) In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ''ξ'' there are infinitely many relatively prime integers ''m'', ''n'' such that \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.see for ex. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal Basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of \R^n under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the pictu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Bilinear Form
In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B that maps every pair (u,v) of elements of the vector space V to the underlying field such that B(u,v)=B(v,u) for every u and v in V. They are also referred to more briefly as just symmetric forms when "bilinear" is understood. Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for ''V''. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2). Given a symmetric bilinear form ''B'', the function is the associated quadratic form on the vector space. Moreover, if the characteristic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definite Bilinear Form
In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical definite noun phrase picks out a unique, familiar, specific referent such as ''the sun'' or ''Australia'', as opposed to indefinite examples like ''an idea'' or ''some fish''. There is considerable variation in the expression of definiteness across languages, and some languages such as Japanese do not generally mark it so that the same expression could be definite in some contexts and indefinite in others. In other languages, such as English, it is usually marked by the selection of determiner (e.g., ''the'' vs ''a''). In still other languages, such as Danish, definiteness is marked morphologically. Definiteness as a grammatical category There are times when a grammatically marked definite NP is not in fact identifiable. For example, ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank–nullity Theorem
The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theorem 2.3 Stating the theorem Let T : V \to W be a linear transformation between two vector spaces where T's domain V is finite dimensional. Then \operatorname(T) ~+~ \operatorname(T) ~=~ \dim V, where \operatorname(T) ~:=~ \dim(\operatorname(T)) \qquad \text \qquad \operatorname(T) ~:=~ \dim(\operatorname (T)). In other words, \dim (\operatorname T) + \dim (\ker T) = \dim (\operatorname T). This theorem can be refined via the splitting lemma to be a statement about an isomorphism of spaces, not just dimensions. Explicitly, since induces an isomorphism from V / \operatorname (T) to \operatorname (T), the existence of a basis for that extends any given basis of \operatorname(T) implies, via the splitting lemma, that \operatorname(T) \oplu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
