Conjugate transpose: Difference between revisions
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(Restoring visibility of task description formulae hidden by under-tested cosmetic edits of 23:08, 14 August 2016) |
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[[Category:Matrices]] |
[[Category:Matrices]] |
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Suppose that a |
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>. |
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::: |
::: <big><math>(M^H)_{ji} = \overline{M_{ij}}</math></big> |
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This means that |
This means that row <big><math>j</math></big>, column <big><math>i</math></big> of the conjugate transpose equals the |
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<br>complex conjugate of |
<br>complex conjugate of row <big><math>i</math></big>, column <big><math>j</math></big> of the original matrix. |
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In the next list, |
In the next list, <big><math>M</math></big> must also be a square matrix. |
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* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: |
* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: <math>M^H = M</math>. |
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* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: |
* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: <math>M^HM = MM^H</math>. |
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* A [[wp:unitary matrix|unitary matrix]] has its [[inverse matrix|inverse]] equal to its conjugate transpose: |
* 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. |
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Also determine if the matrix is a: |
Also determine if the matrix is a: |
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::* |
::* Hermitian matrix, |
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::* |
::* normal matrix, or |
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::* |
::* unitary matrix. |
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;See also: |
;See also: |
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* |
* MathWorld entry: [http://mathworld.wolfram.com/ConjugateTranspose.html conjugate transpose] |
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* |
* MathWorld entry: [http://mathworld.wolfram.com/HermitianMatrix.html Hermitian matrix] |
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* MathWorld entry: [http://mathworld.wolfram.com/NormalMatrix.html normal matrix] |
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* MathWorld entry: [http://mathworld.wolfram.com/UnitaryMatrix.html unitary matrix] |
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<br><br> |
<br><br> |
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