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Complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers. Formal definition Let V be a real vector space. The of is defined by taking the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): :V^ = V\otimes_ \Complex\,. The subscript, \R, on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, V^ is only a real vector space. However, we can make V^ into a complex vector space by defining complex multiplication as follows: :\alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Complex Structure
In mathematics, a complex structure on a real vector space ''V'' is an automorphism of ''V'' that squares to the minus identity, −''I''. Such a structure on ''V'' allows one to define multiplication by complex scalars in a canonical fashion so as to regard ''V'' as a complex vector space. Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure. Complex structures have applications in representation theory as well as in complex geometry where they play an essential role in the definition of almost complex manifolds, by contrast to complex manifolds. The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a linear complex structure. Definition and properties A complex structure on a real vector space ''V'' is a real linear transformation :J :V \to V such that :J^2 = -\mathrm_V. Here means composed with its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension Of Scalars
In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. *f_* M = \operatorname_R(S, M), the coinduced module. *f^* N = N_R, the restriction of scalars. They are related as adjoint functors: :f_! : \text_R \leftrightarrows \text_S : f^* and :f^* : \text_S \leftrightarrows \text_R : f_*. This is related to Shapiro's lemma. Operations Restriction of scalars Throughout this section, let R and S be two rings (they may or may not be commutative, or contain an identity), and let f:R \to S be a homomorphism. Restriction of scalars changes ''S''-modules into ''R''-modules. In algebraic geometry, the term "restriction of scalars" is often used as a synonym for Weil restriction. Definition Suppose that M is a module over S. Then it can be regarded as a module over R where the action of R is given via : \begin M\ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Coordinate Space
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors. The coordinates over any basis of the elements of a real vector space form a ''real coordinate space'' of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension form a ''real coordinate space'' of dimension . These one to one correspondences between vectors, points and coordinate vectors explain the names of ''coordinate space'' and ''coordinate vector''. It allows using geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of calculus in geometry. This approach of geometry was introduced by René Descartes in the 17th century. It is widely used, as it allows loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: * Biquaternions when the coefficients are complex numbers. * Split-biquaternions when the coefficients are split-complex numbers. * Dual quaternions when the coefficients are dual numbers. This article is about the ''ordinary biquaternions'' named by William Rowan Hamilton in 1844 (see ''Proceedings of the Royal Irish Academy'' 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity. The algebra of biquatern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bioctonion
In mathematics, a bioctonion, or complex octonion, is a pair (''p,q'') where ''p'' and ''q'' are biquaternions. The product of two bioctonions is defined using biquaternion multiplication and the biconjugate p → p*: :(p,q)(r,s) = (pr - s^* q,\ sp + q r^*). The bioctonion ''z'' = (''p,q'') has conjugate ''z''* = (''p''*, – ''q''). Then norm ''N''(''z'') of bioctonion ''z'' is ''z z''* = ''p p''* + ''q q''*, which is a complex quadratic form with eight terms. The bioctonion algebra is sometimes introduced as simply the complexification of real octonions, but in abstract algebra it is the result of the Cayley–Dickson construction that begins with the field of complex numbers, the trivial involution, and quadratic form z2. The algebra of bioctonions is an example of an octonion algebra. For any pair of bioctonions ''y'' and ''z'', : N(y z) = N(y) N(z), showing that ''N'' is a quadratic form admitting composition, and hence the bioctonions form a composition algebra. Comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition Algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution called a conjugation: x \mapsto x^*. The quadratic form N(x) = x x^* is called the norm of the algebra. A composition algebra (''A'', ∗, ''N'') is either a division algebra or a split algebra, depending on the existence of a non-zero ''v'' in ''A'' such that ''N''(''v'') = 0, called a null vector. When ''x'' is ''not'' a null vector, the multiplicative inverse of ''x'' is When there is a non-zero null vector, ''N'' is an isotropic quadratic form, and "the algebra splits". Structure theorem Every unital composition algebra over a field can be obtained by repeated application of the Cayley–Dickson construction starting from (if the characteristic of is different from ) or a 2-dimensional composition subalgebra (if ). The poss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and which allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate-linear Map
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y \in V and every complex number s, where \overline denotes the complex conjugate of s. Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity. Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces. Definitions and characterizations A function is called or if it is additive and conjugate homogeneous. An o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicomplex Number
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z + v w). Then the bicomplex norm is given by :(w,z)^* (w,z) = (w, -z)(w,z) = (w^2 + z^2, 0), a quadratic form in the first component. The bicomplex numbers form a commutative algebra over C of dimension two, which is isomorphic to the direct sum of algebras . The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers: a verification of this property of the quadratic form of a product refers to the Brahmagupta–Fibonacci identity. This property of the quadratic form of a bicomplex number indicates that these numbers form a composition algebra. In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on \mathbb with no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octonion
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 friend Wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
