Complex conjugate of a vector space information

In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} that...

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite 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...

mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle...

the conjugate transpose, also known as the Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle \mathbf {A} } is...

called vector axioms. Real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars: real coordinate space or...

In mathematics, a function f : V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if f ( x...

the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: A  is Hermitian ⟺ a i j = a j i ¯ {\displaystyle A{\text{...

and a vector of the original vector space. A linear map A from a vector space of dimension n into a vector space of dimension m maps a column vector x =...

Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces...

In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle...

complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained...

mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype...

product of a complex vector a ⋅ a = a H a {\displaystyle \mathbf {a} \cdot \mathbf {a} =\mathbf {a} ^{\mathsf {H}}\mathbf {a} } , involving the conjugate transpose...

is a group and Π is a representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate...

inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse. For two vector quaternions...

(y); Conjugate homogeneous: f (c x) = c f (x). The vector space of all continuous antilinear functions on H is called the anti-dual space or complex conjugate...

In mathematics, a complex structure on a real vector space V {\displaystyle V} is an automorphism of V {\displaystyle V} that squares to the minus identity...

functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows...

{p} _{n}\}} is a set of n {\displaystyle n} mutually conjugate vectors with respect to A {\displaystyle \mathbf {A} } , i.e. p i T A p j = 0 {\displaystyle...

mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms...

areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function...

elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected...

a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over...

equals the inverse. Over a complex vector space, one often works with sesquilinear forms (conjugate-linear in one argument) instead of bilinear forms. The...

the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices...

complex structures. Consider an N-manifold M. The tangent bundle of M, which will be denoted T, is the vector bundle over M whose fibers consist of all...