complex conjugate (original) (raw)

1 Definition

1.1 Scalar Complex Conjugate

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=(ai⁢j) be a n×m matrix with complex entries. Then the complex conjugate of A is the matrixA¯=(ai⁢j¯). In particular, ifv=(v1,…,vn) is a complex row/column vector, thenv¯=(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. u⁢v¯=(u¯)⁢(v¯)
2. 1. u+v¯=u¯+v¯
3. 1. (u¯)-1=u-1¯
4. 1. (u¯)¯=u
5. 1. If v≠0, then (uv)¯=u¯/v¯
7. 1. If z is written in polar form as z=r⁢ei⁢ϕ, thenz¯=r⁢e-i⁢ϕ.

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. AT¯=(A¯)T
2. 1. A⁢v¯=A¯⁢v¯, and v⁢A¯=v¯⁢A¯. (Here we assume that A and v arecompatible size.)

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

1. 1. trace⁡A¯=(trace⁡A)¯
2. 1. det⁡A¯=(det⁡A)¯
3. 1. (A¯)-1=A-1¯