This article covers complex number conjugation. For conjugation in groups see conjugation (group theory) .
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 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:
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