complex conjugate

1 Definition

1.1 Scalar Complex Conjugate

Let z be a complex numberMathworldPlanetmathPlanetmath with real partMathworldPlanetmath a and imaginary part b,


Then the complex conjugateMathworldPlanetmath of z is


Complex conjugation represents a reflectionMathworldPlanetmath about the real axis on the Argand diagramMathworldPlanetmath representing a complex number.

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


where Ψ* is the complex conjugate of a wave .

1.2 Matrix Complex Conjugate

Let A=(aij) be a n×m matrix with complex entries. Then the complex conjugate of A is the matrix A¯=(aij¯). In particular, if v=(v1,,vn) is a complex row/column vectorMathworldPlanetmath, then v¯=(v1¯,,vn¯).

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.


  2. 2.


  3. 3.


  4. 4.


  5. 5.

    If v0, then (uv)¯=u¯/v¯

  6. 6.

    Let u=a+bi. Then u¯u=uu¯=a2+b20 (the complex modulusMathworldPlanetmath).

  7. 7.

    If z is written in polar form as z=reiϕ, then z¯=re-iϕ.

2.2 Matrix and Vector Properties

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


  1. 1.


  2. 2.

    Av¯=A¯v¯, and vA¯=v¯A¯. (Here we assume that A and v are compatible size.)

Now assume further that A is a complex square matrixMathworldPlanetmath, then

  1. 1.


  2. 2.


  3. 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