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

$$\overline{z}=a-bi$$

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

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\Psi }}^{*}\mathrm{\Psi }𝑑x=1$$

where ${\mathrm{\Psi }}^{*}$ 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 $v=(v^1,\mathrm{\dots },v^n)$ is a complex row/column vector, then $\overline{v}=(\overline{v^1},\mathrm{\dots },\overline{v^n})$.

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

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

3. ${\left(\overline{u}\right)}^{-1}=\overline{u^{-1}}$

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

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

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

7. If $z$ is written in polar form as $z=r{e}^{i\varphi }$, then $\overline{z}=r{e}^{-i\varphi }$.

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. $\overline{A^T}={\left(\overline{A}\right)}^T$

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. $\mathrm{trace}\overline{A}=\overline{(\mathrm{trace}A)}$

2. $det\overline{A}=\overline{(detA)}$

3. ${\left(\overline{A}\right)}^{-1}=\overline{A^{-1}}$

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