Conjugation (math)

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The green pointer in the upper part of the picture describes the complex number in the complex number plane ( Gaussian number plane ). The complex conjugate is created by reflection on the x-axis (lower green pointer). The blue lines should indicate the real and imaginary parts.

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 .


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.


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.


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:


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

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