# complex conjugate

## 1 Definition

### 1.1 Scalar Complex Conjugate

Let $z$ be a complex number with real part $a$ and imaginary part $b$,

 $z=a+bi$

Then the complex conjugate of $z$ is

 $\bar{z}=a-bi$

Complex conjugation represents a reflection about the real axis on the Argand diagram representing a complex number.

Sometimes a star ($*$) is used instead of an overline, e.g. in physics you might see

 $\int_{-\infty}^{\infty}\Psi^{*}\Psi dx=1$

where $\Psi^{*}$ is the complex conjugate of a wave .

### 1.2 Matrix Complex Conjugate

Let $A=(a_{ij})$ be a $n\times m$ matrix with complex entries. Then the complex conjugate of $A$ is the matrix $\overline{A}=(\overline{a_{ij}})$. In particular, if $v=(v^{1},\ldots,v^{n})$ is a complex row/column vector, then $\overline{v}=(\overline{v^{1}},\ldots,\overline{v^{n}})$.

Hence, the matrix complex conjugate is what we would expect: the same matrix with all of its scalar components conjugated.

## 2 Properties of the Complex Conjugate

### 2.1 Scalar Properties

If $u,v$ are complex numbers, then

1. 1.

$\overline{uv}=(\overline{u})(\overline{v})$

2. 2.

$\overline{u+v}=\overline{u}+\overline{v}$

3. 3.

$\big{(}\overline{u}\big{)}^{-1}=\overline{u^{-1}}$

4. 4.

$\overline{(\overline{u})}=u$

5. 5.

If $v\neq 0$, then $\overline{(\frac{u}{v})}={\overline{u}}/{\overline{v}}$

6. 6.

Let $u=a+bi$. Then $\overline{u}u=u\overline{u}=a^{2}+b^{2}\geq 0$ (the complex modulus).

7. 7.

If $z$ is written in polar form as $z=re^{i\phi}$, then $\overline{z}=re^{-i\phi}$.

### 2.2 Matrix and Vector Properties

Let $A$ be a matrix with complex entries, and let $v$ be a complex row/column vector.

Then

1. 1.

$\overline{A^{T}}=\big{(}\overline{A}\big{)}^{T}$

2. 2.

$\overline{Av}=\overline{A}\overline{v}$, and $\overline{vA}=\overline{v}\overline{A}$. (Here we assume that $A$ and $v$ are compatible size.)

Now assume further that $A$ is a complex square matrix, then

1. 1.

$\operatorname{trace}\overline{A}=\overline{(\operatorname{trace}\ A)}$

2. 2.

$\det\overline{A}=\overline{(\det A)}$

3. 3.

$\big{(}\overline{A}\big{)}^{-1}=\overline{A^{-1}}$

 Title complex conjugate Canonical name ComplexConjugate Date of creation 2013-03-22 12:12:03 Last modified on 2013-03-22 12:12:03 Owner akrowne (2) Last modified by akrowne (2) Numerical id 11 Author akrowne (2) Entry type Definition Classification msc 12D99 Classification msc 30-00 Classification msc 32-00 Related topic Complex Related topic ModulusOfComplexNumber Related topic AlgebraicConjugates Related topic TriangleInequalityOfComplexNumbers Related topic Antiholomorphic2 Defines complex conjugation Defines matrix complex conjugate