Complex Conjugate - Generalizations

Generalizations

The other planar real algebras, dual numbers, and split-complex numbers are also explicated by use of complex conjugation.

For matrices of complex numbers, where means the element-by-element conjugation of . Contrast this to the property, where means the conjugate transpose of .

Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C*-algebras.

One may also define a conjugation for quaternions and coquaternions: the conjugate of is .

Note that all these generalizations are multiplicative only if the factors are reversed:

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 over the complex numbers. In this context, any antilinear map that satisfies

1. , where and is the identity map on ,
2. for all, and
3. for all, ,

is called a complex conjugation, or a real structure. As the involution is antilinear, it cannot be the identity map on . Of course, is a -linear transformation of, if one notes that every complex space V has a real form obtained by taking the same vectors 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 . One example of this notion is the conjugate transpose operation of complex matrices defined above. It should be remarked that on generic complex vector spaces there is no canonical notion of complex conjugation.