PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: Very high
complex conjugate (Definition)

Definition

Scalar Complex Conjugate

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

$\displaystyle z = a+bi $

Then the complex conjugate of $ z$ is

$\displaystyle \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

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

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

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.

Properties of the Complex Conjugate

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. $ \big(\overline{u}\big)^{-1} = \overline{u^{-1}}$
  4. $ \overline{(\overline{u})} = u$
  5. If $ v\neq 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\phi}$, then $ \overline{z}=re^{-i\phi}$.

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}=\big(\overline{A}\big)^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. $ \operatorname{trace}\overline{A} = \overline{(\operatorname{trace}\ A)}$
  2. $ \det \overline{A} = \overline{(\det A)}$
  3. $ \big(\overline{A}\big)^{-1} = \overline{A^{-1}}$



"complex conjugate" is owned by akrowne.
(view preamble)

View style:

See Also: complex, modulus of complex number, algebraic conjugates

Also defines:  complex conjugation, matrix complex conjugate

Attachments:
conjugated roots of equation (Topic) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: square matrix, size, compatible, polar form, complex modulus, components, scalar, row vector, complex, matrix, star, Argand diagram, real axis, reflection, represents, imaginary part, real part, complex number
There are 46 references to this entry.

This is version 7 of complex conjugate, born on 2002-01-21, modified 2004-02-25.
Object id is 1508, canonical name is ComplexConjugate.
Accessed 23331 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 30-00 (Functions of a complex variable :: General reference works )
 32-00 (Several complex variables and analytic spaces :: General reference works )
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)