Conjugate transpose: Difference between revisions

Revision as of 17:48, 3 August 2019

Revision as of 16:29, 14 September 2019

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* <math>\boldsymbol{A}</math> is [[invertible matrix|invertible]] [[if and only if]] <math>\boldsymbol{A}^\mathrm{H}</math> is invertible, and in that case <math>(\boldsymbol{A}^\mathrm{H})^{-1} = (\boldsymbol{A}^{-1})^{\mathrm{H}}</math>.

* The [[eigenvalue]]s of <math>\boldsymbol{A}^\mathrm{H}</math> are the complex conjugates of the [[eigenvalue]]s of <math>\boldsymbol{A}</math>.

* <math>\langle \boldsymbol{A} x,y \ranglerangle_m = \langle x, \boldsymbol{A}^\mathrm{H} y\ranglerangle_n </math> for any ''m''-by-''n'' matrix <math>\boldsymbol{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\ranglerangle_m</math> denotes the standard complex [[inner product]] on <math> \mathbb{C}^m </math>, and similarly for <math> \mathbb{C}^n langle\cdot,\cdot\rangle_n</math>.

==Generalizations==