Conjugate transpose: Difference between revisions
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[[Category:Matrices]] |
[[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]] <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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: <math> (M^H)_{ji} = \overline{M_{ij}} </math> |
::: <big><big><math> (M^H)_{ji} = \overline{M_{ij}} </math></big></big> |
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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. |
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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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⚫ | * 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. |
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Given some matrix of complex numbers, find its conjugate transpose. |
Given some matrix of complex numbers, find its conjugate transpose. |
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Also determine if |
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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