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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 sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
Geometric representation of z and its conjugate z in the complex plane. The complex conjugate of the complex number z = x + yi is defined as ¯ =. [11] It is also denoted by some authors by . Geometrically, z is the "reflection" of z about the real axis.
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. [1]
The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an matrix made up of complex numbers. For an explanation of the notation used here, we begin by representing complex numbers e i θ {\displaystyle e^{i\theta }} as the rotation matrix, that is,
If is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π (X) is the conjugate of −π(X*) for all X in g. For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
The map f C commutes with conjugation and so maps the real subspace of V C to the real subspace of W C (via the map f). Moreover, a complex linear map g : V C → W C is the complexification of a real linear map if and only if it commutes with conjugation. As an example consider a linear transformation from R n to R m thought of as an m×n matrix.
The letter stands for a vector in , is a complex number, and ¯ denotes the complex conjugate of . [1] More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure (different multiplication by ).
Given a complex Lie algebra , its conjugate ¯ is a complex Lie algebra with the same underlying real vector space but with = acting as instead. [1] As a real Lie algebra, a complex Lie algebra g {\displaystyle {\mathfrak {g}}} is trivially isomorphic to its conjugate.