# Conjugation (math)

In mathematics , a **complex conjugation is** the mapping

with in the body of complex numbers . It is a body automorphism of , i.e. compatible with addition and multiplication:

- .

The number is referred to as the too **complex conjugate** or **conjugate complex** number or **conjugate for** short .

## General

In the exponential form is the conjugate of the number

the number

With the same amount it has the opposite sign of . One can identify the conjugation in the complex number plane as the *reflection on the real axis* . In particular, the real numbers are mapped back to themselves during conjugation.

## Spellings

An alternative way of writing is that especially in the physics , more in the quantum mechanics , in use is (with is to referred conjugate wave function). This notation is also used for adjoint matrices , for which the notation is again used in quantum mechanics .

## Calculation rules

The following applies to all complex numbers :

- For

- applies in general to every holomorphic function whose restriction to the real axis is real-valued.

## application

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

- To with is

- the multiplicative inverse.

- For dividing two complex numbers we get:

- or more detailed:

## Complex conjugation for matrices

The 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 a Hermitian transposition. For matrices in Euclidean space it still applies that the Hermitian transposed matrix is identical to the adjoint matrix .

Since the operation is a simple extension of the conjugation of matrix elements to matrices, the complex conjugate of a matrix is often also marked with an overline. A simple calculation example:

## generalization

In abstract algebra this term is expanded as follows:

Two over algebraic elements of a field extension are called *conjugate to each other* if they have the same minimal polynomial over . The zeros of the minimal polynomial of in are called “conjugates of (in )”. Everyone - automorphism of (ie a -Automorphismus, the point-holds) is off to one of its conjugates.

Similarly, one defines the conjugation of elements and ideals with respect to a ring expansion.

## Individual evidence

- ↑ Gerhard Merziger, Thomas Wirth: Repetition of higher mathematics . 5th edition. Binomi, 2006, ISBN 978-3-923923-33-5 , pp. 98 .
- ^ Bronstein, Semendjajew, Musiol, Mühlig: Taschenbuch der Mathematik, Verlag Harri Deutsch, page 36
- ↑ T. Arens, F. Hettlich, Ch.Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel: Mathematik, Spektrum Akademischer Verlag, pages 125 to 127