|
[[Category:Matrices]]
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><big><math>(M^H)_{ji} = \overline{M_{ij}}</math></big></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: <big><math>M^H = M.</math></big>.
* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: <big><math>M^HM = MM^H.</math></big>.
* A [[wp:unitary matrix|unitary matrix]] has its [[inverse matrix|inverse]] equal to its conjugate transpose: <big><math>M^H = M^{-1}.</math></big>. <br> This is true when: <br> [[wikt:iff|'''iff''']] <math>M^HM = I_n</math> and <br> [[wikt:iff|'''iff''']] <big><math>MM^H = I_n,</math></big> , where <big><math>I_n</math></big> is the identity matrix.
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]
<br><br>
| 