Suppose that a [[matrix]] <big><math>M</math></big> contains [[Arithmetic/Complex|complex numbers]]. Then the [[wp:conjugate transpose|conjugate transpose]] of <math>M</math> is a matrix <math>M^H</math> containing the [[complex conjugate]]s of the [[matrix transposition]] of <math>M</math>.

::: <big><math>(M^H)_{ji} = \overline{M_{ij}}</math></big>


This means that   row <big><math>j</math></big>, column <big><math>i</math></big> of the conjugate transpose equals the 
<br>complex conjugate of row <big><math>i</math></big>, column <big><math>j</math></big> of the original matrix.


In the next list, <big><math>M</math></big> must also be a square matrix.

* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: <math>M^H = M</math>.
* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: <math>M^HM = MM^H</math>.
* A [[wp:unitary matrix|unitary matrix]] has its [[inverse matrix|inverse]] equal to its conjugate transpose: <math>M^H = M^{-1}</math>. <br> This is true [[wikt:iff|iff]] <math>M^HM = I_n</math> and iff <math>MM^H = I_n</math>, where <math>I_n</math> is the identity matrix.


<br>
;Task:
Given some matrix of complex numbers, find its conjugate transpose. 

Also determine if the matrix is a:
::* Hermitian matrix,
::* normal matrix,  or 
::* unitary matrix.


;See also:
* MathWorld entry: [http://mathworld.wolfram.com/ConjugateTranspose.html conjugate transpose]
* MathWorld entry: [http://mathworld.wolfram.com/HermitianMatrix.html Hermitian matrix]
* MathWorld entry: [http://mathworld.wolfram.com/NormalMatrix.html normal matrix]
* MathWorld entry: [http://mathworld.wolfram.com/UnitaryMatrix.html unitary matrix]
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