Conjugate transpose: Difference between revisions
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[[Category:Matrices]] |
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Suppose that a [[matrix]] <math>M</math> 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>. |
Suppose that a [[matrix]] <math>M</math> 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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: <math>(M^H)_{ji} = \overline{M_{ij}}</math> |
: <math> (M^H)_{ji} = \overline{M_{ij}} </math> |
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This means that row <math>j</math>, column <math>i</math> of the conjugate transpose equals the complex conjugate of row <math>i</math>, column <math>j</math> of the original matrix. |
This means that row <math>j</math>, column <math>i</math> of the conjugate transpose equals the complex conjugate of row <math>i</math>, column <math>j</math> of the original matrix. |
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* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: <math>M^H = M</math>. |
* 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: <math>M^HM = MM^H</math>. |
* 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: <math>M^H = M^{-1}</math>. 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. |
* 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 it is a Hermitian matrix, normal matrix, or a unitary matrix. |
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;See also: |
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* MathWorld: [http://mathworld.wolfram.com/ConjugateTranspose.html conjugate transpose], [http://mathworld.wolfram.com/HermitianMatrix.html Hermitian matrix], [http://mathworld.wolfram.com/NormalMatrix.html normal matrix], [http://mathworld.wolfram.com/UnitaryMatrix.html unitary matrix] |
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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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=={{header|Ada}}== |
=={{header|Ada}}== |
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