# Conjugation (math)

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.${\ displaystyle z = a + b \ mathrm {i}}$${\ displaystyle {\ bar {z}} = from \ mathrm {i}}$

In mathematics , a complex conjugation is the mapping

${\ displaystyle \ mathbb {C} \ to \ mathbb {C}, \ quad z = a + b \ cdot \ mathrm {i} \; \; \ mapsto \; \; {\ bar {z}} = ab \ cdot \ mathrm {i}}$

with in the body of complex numbers . It is a body automorphism of , i.e. compatible with addition and multiplication: ${\ displaystyle a, b \ in \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$

${\ displaystyle {\ overline {y + z}} = {\ bar {y}} + {\ bar {z}}, \ quad {\ overline {y \ cdot z}} = {\ bar {y}} \ cdot {\ bar {z}}}$.

The number is referred to as the too complex conjugate or conjugate complex number or conjugate for short . ${\ displaystyle {\ bar {z}} = from \ cdot \ mathrm {i}}$${\ displaystyle z = a + b \ cdot \ mathrm {i}}$

## General

In the exponential form is the conjugate of the number

${\ displaystyle \! \ z = re ^ {\ mathrm {i} \ varphi} = r (\ cos \ varphi + \ mathrm {i} \ sin \ varphi)}$

the number

${\ displaystyle {\ bar {z}} = re ^ {- \ mathrm {i} \ varphi} = r (\ cos \ varphi - \ mathrm {i} \ sin \ varphi).}$

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. ${\ displaystyle z}$

## 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 . ${\ displaystyle {\ overline {z}}}$${\ displaystyle z ^ {*}}$${\ displaystyle \ psi ^ {*} ({\ vec {x}}, t)}$${\ displaystyle \ psi ({\ vec {x}}, t)}$ ${\ displaystyle A ^ {*}: = {\ overline {A}} ^ {T}}$${\ displaystyle A ^ {\ dagger}}$

## Calculation rules

The following applies to all complex numbers : ${\ displaystyle z_ {1}, z_ {2}, z = a + b \, \ mathrm {i} \ in \ mathbb {C}}$

• ${\ displaystyle a = \ mathrm {Re} (z) = {\ frac {1} {2}} (z + {\ overline {z}})}$
• ${\ displaystyle b = \ mathrm {Im} (z) = {\ frac {1} {2 \ mathrm {i}}} (z - {\ overline {z}})}$
• ${\ displaystyle {\ overline {\ overline {z}}} = z}$
• ${\ displaystyle z \ in \ mathbb {R} \ iff {\ overline {z}} = z}$
• ${\ displaystyle z \ cdot {\ overline {z}} = | z | ^ {2} = a ^ {2} + b ^ {2}}$
• ${\ displaystyle {\ overline {z_ {1} + z_ {2}}} = {\ overline {z}} _ {1} + {\ overline {z}} _ {2}}$
• ${\ displaystyle {\ overline {z_ {1} \ cdot z_ {2}}} = {\ overline {z}} _ {1} \ cdot {\ overline {z}} _ {2}}$
• ${\ displaystyle {\ overline {\ left ({\ frac {z_ {1}} {z_ {2}}} \ right)}} = {\ frac {{\ overline {z}} _ {1}} {{ \ overline {z}} _ {2}}}}$
• ${\ displaystyle | z | = | {\ overline {z}} |}$
• ${\ displaystyle \ exp ({\ overline {z}}) = {\ overline {\ exp (z)}}}$
• ${\ displaystyle \ log ({\ overline {z}}) = {\ overline {\ log (z)}}}$ For ${\ displaystyle z \ neq 0}$
• ${\ displaystyle {\ overline {\ varphi (z)}} = \ varphi ({\ overline {z}})}$applies in general to every holomorphic function whose restriction to the real axis is real-valued.${\ displaystyle \! \ \ varphi}$

## application

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

• To with is${\ displaystyle z \ in \ mathbb {C}}$${\ displaystyle z \ neq 0}$
${\ displaystyle z ^ {- 1} = {\ frac {1} {z}} = {\ frac {1} {z}} {\ frac {\ bar {z}} {\ bar {z}}} = {\ frac {\ bar {z}} {| z | ^ {2}}}}$
the multiplicative inverse.
• For dividing two complex numbers we get:
${\ displaystyle {y \ over z} = {y \ over z} {\ frac {\ bar {z}} {\ bar {z}}} = {\ frac {y {\ bar {z}}} {| z | ^ {2}}}}$
or more detailed:
${\ displaystyle {\ frac {a + b \ mathrm {i}} {c + d \ mathrm {i}}} = {\ frac {ac + bd} {c ^ {2} + d ^ {2}}} + {\ frac {bc-ad} {c ^ {2} + d ^ {2}}} \, \ mathrm {i}.}$

## 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:

${\ displaystyle A = {\ begin {pmatrix} 2 & \ mathrm {i} & 3 + \ mathrm {i} \\ - \ mathrm {i} & 5 + 3 \ mathrm {i} & 5 \ mathrm {i} \\\ end { pmatrix}} \ Leftrightarrow \, {\ overline {A}} = {\ begin {pmatrix} 2 & - \ mathrm {i} & 3- \ mathrm {i} \\\ mathrm {i} & 5-3 \ mathrm {i} & -5 \ mathrm {i} \\\ end {pmatrix}}}$

## 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. ${\ displaystyle K}$ ${\ displaystyle L / K}$${\ displaystyle K}$${\ displaystyle a}$${\ displaystyle L}$${\ displaystyle a}$${\ displaystyle L}$${\ displaystyle K}$${\ displaystyle L}$${\ displaystyle L}$${\ displaystyle K}$${\ displaystyle a}$

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