complex conjugate
1 Definition
1.1 Scalar Complex Conjugate
Let be a complex number with real part
and imaginary part ,
Then the complex conjugate of is
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
where is the complex conjugate of a wave .
1.2 Matrix Complex Conjugate
Let be a matrix with complexentries. Then the complex conjugate of is the matrix. In particular, if is a complex row/column vector, then.
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 are complex numbers, then
- 1.
- 2.
- 3.
- 4.
- 5.
If , then
- 6.
Let . Then (the complex modulus
).
- 7.
If is written in polar form as , then.
2.2 Matrix and Vector Properties
Let be a matrix with complex entries, and let be a complex row/column vector.
Then
- 1.
- 2.
, and . (Here we assume that and arecompatible size.)
Now assume further that is a complex square matrix, then
- 1.
- 2.
- 3.
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 |