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Conjugate transpose: Difference between revisions
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[[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
In the next list, <math>M</math> must also be a square matrix.▼
<br>complex conjugate of row <big><math> i, </math></big> column <big><math> j </math></big> of the original matrix.
* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: <math>M^H = M</math>.▼
▲In the next list, <big><math> M </math></big> must also be a square matrix.
* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: <math>M^HM = MM^H</math>.▼
* 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.▼
▲* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: <big><math> M^H = M. </math>
▲* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: <big><math> M^HM = MM^H. </math>
▲* A [[wp:unitary matrix|unitary matrix]] has its [[inverse matrix|inverse]] equal to its conjugate transpose: <big><math> M^H = M^{-1}. </math>
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Given some matrix of complex numbers, find its conjugate transpose.
Also determine if
::* Hermitian matrix,
::* normal matrix, or
::* unitary matrix.
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