Editing Complex conjugate (section)

==Generalizations==
The other planar real unital algebras, [[dual numbers]], and [[split-complex number]]s are also analyzed using complex conjugation.

For matrices of complex numbers, <math display="inline">\overline{\mathbf{AB}} = \left(\overline{\mathbf{A}}\right) \left(\overline{\mathbf{B}}\right),</math> where <math display="inline">\overline{\mathbf{A}}</math> represents the element-by-element conjugation of <math>\mathbf{A}.</math><ref>Arfken, ''Mathematical Methods for Physicists'', 1985, pg. 201</ref> Contrast this to the property <math display="inline">\left(\mathbf{AB}\right)^*=\mathbf{B}^* \mathbf{A}^*,</math> where <math display="inline">\mathbf{A}^*</math> represents the [[conjugate transpose]] of <math display="inline">\mathbf{A}.</math>

Taking the [[conjugate transpose]] (or adjoint) of complex [[Matrix (mathematics)|matrices]] generalizes complex conjugation. Even more general is the concept of [[adjoint operator]] for operators on (possibly infinite-dimensional) complex [[Hilbert space]]s. All this is subsumed by the *-operations of [[C*-algebra]]s.

One may also define a conjugation for [[quaternion]]s and [[split-quaternion]]s: the conjugate of <math display="inline">a + bi + cj + dk</math> is <math display="inline">a - bi - cj - dk.</math>

All these generalizations are multiplicative only if the factors are reversed:
<math display="block">{\left(zw\right)}^* = w^* z^*.</math>

Since the multiplication of planar real algebras is [[commutative]], this reversal is not needed there.

There is also an abstract notion of conjugation for [[vector spaces]] <math display="inline">V</math> over the [[complex number]]s. In this context, any [[antilinear map]] <math display="inline">\varphi: V \to V</math> that satisfies

# <math>\varphi^2 = \operatorname{id}_V\,,</math> where <math>\varphi^2 = \varphi \circ \varphi</math> and <math>\operatorname{id}_V</math> is the [[identity map]] on <math>V,</math>
# <math>\varphi(zv) = \overline{z} \varphi(v)</math> for all <math>v \in V, z \in \Complex,</math> and
# <math>\varphi\left(v_1 + v_2\right) = \varphi\left(v_1\right) + \varphi\left(v_2\right)\,</math> for all <math>v_1, v_2 \in V,</math>

is called a {{em|complex conjugation}}, or a [[real structure]]. As the involution <math>\varphi</math> is [[antilinear]], it cannot be the identity map on <math>V.</math>

Of course, <math display="inline">\varphi</math> is a <math display="inline">\R</math>-linear transformation of <math display="inline">V,</math> if one notes that every complex space <math>V</math> has a real form obtained by taking the same [[vector (mathematics and physics)|vector]]s as in the original space and restricting the scalars to be real. The above properties actually define a real structure on the complex vector space <math>V.</math><ref>Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988, p. 29</ref> 

One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is no {{em|[[Canonical form|canonical]]}} notion of complex conjugation.