Editing Conjugate transpose (section)

==Properties==
* <math>(\mathbf{A} + \boldsymbol{B})^\mathrm{H} = \mathbf{A}^\mathrm{H} + \boldsymbol{B}^\mathrm{H}</math> for any two matrices <math>\mathbf{A}</math> and <math>\boldsymbol{B}</math> of the same dimensions.
* <math>(z\mathbf{A})^\mathrm{H} = \overline{z} \mathbf{A}^\mathrm{H}</math> for any complex number <math>z</math> and any <math>m \times n</math> matrix <math>\mathbf{A}</math>.
* <math>(\mathbf{A}\boldsymbol{B})^\mathrm{H} = \boldsymbol{B}^\mathrm{H} \mathbf{A}^\mathrm{H}</math> for any <math>m \times n</math> matrix <math>\mathbf{A}</math> and any <math>n \times p</math> matrix <math>\boldsymbol{B}</math>. Note that the order of the factors is reversed.<ref name=":1" />
* <math>\left(\mathbf{A}^\mathrm{H}\right)^\mathrm{H} = \mathbf{A}</math> for any <math>m \times n</math> matrix <math>\mathbf{A}</math>, i.e. Hermitian transposition is an [[Involution (mathematics)|involution]].
* If <math>\mathbf{A}</math> is a square matrix, then <math>\det\left(\mathbf{A}^\mathrm{H}\right) = \overline{\det\left(\mathbf{A}\right)}</math> where <math>\operatorname{det}(A)</math> denotes the [[determinant]] of <math>\mathbf{A}</math> .
* If <math>\mathbf{A}</math> is a square matrix, then <math>\operatorname{tr}\left(\mathbf{A}^\mathrm{H}\right) = \overline{\operatorname{tr}(\mathbf{A})}</math> where <math>\operatorname{tr}(A)</math> denotes the [[trace (matrix)|trace]] of <math>\mathbf{A}</math>.
* <math>\mathbf{A}</math> is [[invertible matrix|invertible]] [[if and only if]] <math>\mathbf{A}^\mathrm{H}</math> is invertible, and in that case <math>\left(\mathbf{A}^\mathrm{H}\right)^{-1} = \left(\mathbf{A}^{-1}\right)^{\mathrm{H}}</math>.
* The [[eigenvalue]]s of <math>\mathbf{A}^\mathrm{H}</math> are the complex conjugates of the [[eigenvalue]]s of <math>\mathbf{A}</math>.
* <math>\left\langle \mathbf{A} x,y \right\rangle_m = \left\langle x, \mathbf{A}^\mathrm{H} y\right\rangle_n </math> for any <math>m \times n</math> matrix <math>\mathbf{A}</math>, any vector in <math>x \in \mathbb{C}^n </math> and any vector <math>y \in \mathbb{C}^m </math>. Here, <math>\langle\cdot,\cdot\rangle_m</math> denotes the standard complex [[inner product]] on <math> \mathbb{C}^m </math>, and similarly for <math>\langle\cdot,\cdot\rangle_n</math>.