Complex conjugate

In mathematics called complex conjugation, the mapping

In the field of complex numbers. She is a Körperautomorphismus from, so with the addition and multiplication compatible:

Thus, the number has to conjugate the same real part but the opposite sign in the imaginary part.

In the exponential form is the conjugate of the number

The number

So she has an unchanged amount opposite in sign angle of, for Man, the conjugation in the complex plane that is identified as the reflection in the real axis. Especially in the conjugation, the real numbers are mapped back onto itself exactly. An alternative way of writing is, which is mainly in physics, specifically in quantum mechanics, common ( with the conjugate wave function referred to ). However, this notation is also used in the adjoint matrices for which in quantum mechanics again the notation is in use.

Calculation rules

For all complex numbers shall:

• Applies generally to any holomorphic function whose restriction is real-valued on the real axis.

Application

With the help of the inverse conjugation and also the quotient of complex numbers can be conveniently specified:

• It should be with

• For the division of two complex numbers we get:

Complex conjugation with matrices

The complex conjugate of a matrix is ​​the matrix whose components are the complex conjugate components of the original matrix. The transposition of a previously complex conjugate matrix is called Hermitian transposition. Continues to apply to matrices on the Euclidean space that the Hermitian transpose matrix is identical to the adjoint matrix.

Since the operation is a simple extension of conjugation of matrix elements of matrices is the complex conjugate of a matrix is often also characterized by a top stroke. A simple calculation:

Generalization

In abstract algebra, this term is extended as follows:

Two on algebraic elements of a field extension called conjugate to each other if they have the same minimal polynomial over. The zeros of the minimal polynomial of in are called " conjugate of ( in) '. Each automorphism of (ie, a - automorphism which holds pointwise ) reflects on one of its conjugates.

Analogously one defines conjugacy of elements and ideals with respect to a ring expansion.