Conjugate transpose: Difference between revisions

Suppose that a &nbsp; [[matrix]] &nbsp; <big><math> M </math></big> &nbsp; contains &nbsp; [[Arithmetic/Complex|complex numbers]]. &nbsp; Then the &nbsp; [[wp:conjugate transpose|conjugate transpose]] &nbsp; of &nbsp; <math> M </math> &nbsp; is a matrix &nbsp; <math> M^H </math> &nbsp; containing the &nbsp; [[complex conjugate]]s &nbsp; of the [[matrix transposition]] &nbsp; of &nbsp; <math> M. </math>

::: <big><big><math> (M^H)_{ji} = \overline{M_{ij}} </math></big></big>

This means that &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; row &nbsp; <big><math> j,</math></big> &nbsp; column &nbsp; <big><math> i </math></big> &nbsp; of the conjugate transpose equals the

<br>complex conjugate of &nbsp; row &nbsp; <big><math> i, </math></big> &nbsp; column &nbsp; <big><math> j </math></big> &nbsp; of the original matrix.

In the next list, &nbsp; <big><math> M </math></big> &nbsp; must also be a square matrix.

* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: &nbsp; <big><math> M^H = M. </math></big>

* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: &nbsp; <big><math> M^HM = MM^H. </math></big>

* A [[wp:unitary matrix|unitary matrix]] has its [[inverse matrix|inverse]] equal to its conjugate transpose: &nbsp; <big><math> M^H = M^{-1}. </math></big> <br> This is true when: <br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; [[wikt:iff|'''iff''']] &nbsp; <math> M^HM = I_n </math> &nbsp; and <br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; [[wikt:iff|'''iff''']] &nbsp; <big><math> MM^H = I_n,</math></big> &nbsp; where &nbsp; <big><math> I_n </math></big> &nbsp; is the identity matrix.