1.1 Scalar Complex Conjugate
Then the complex conjugate of is
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 complex entries. 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
If , then
Let . Then (the complex modulus).
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.
, and . (Here we assume that and are compatible size.)
Now assume further that is a complex square matrix, then
|Date of creation||2013-03-22 12:12:03|
|Last modified on||2013-03-22 12:12:03|
|Last modified by||akrowne (2)|
|Defines||matrix complex conjugate|