34 lines
1.7 KiB
Plaintext
34 lines
1.7 KiB
Plaintext
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 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 row <big><math>i</math></big>, column <big><math>j</math></big> of the original matrix.
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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: <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>.
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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>. <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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<br>
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;Task:
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Given some matrix of complex numbers, find its conjugate transpose.
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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:
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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>
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