Conjugate transpose: Difference between revisions

Revision as of 08:21, 7 March 2021

Revision as of 16:30, 26 March 2021

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where the subscriptssubscript denote<math>ij</math> denotes the <math>(i,j)</math>-th entry, for <math>1 \le i \le n</math> and <math>1 \le j \le m</math>, and the overbar denotes a scalar complex conjugate.

This definition can also be written as<ref name=":2" />

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==Basic remarks==

A square matrix <math>\boldsymbol{A}</math> with entries <math>a_{ij}</math> is called

* [[hermitian matrix|Hermitian]] or [[self-adjoint_operator|self-adjoint]] if <math>\boldsymbol{A}=\boldsymbol{A}^\mathrm{H}</math>; i.e., <math>a_{ij} = \overline{a_{ji}}</math>&nbsp;.

* [[skew-Hermitian matrix|Skew Hermitian]] or antihermitian if <math>\boldsymbol{A}=-\boldsymbol{A}^\mathrm{H}</math>; i.e., <math>a_{ij} = -\overline{a_{ji}}</math>&nbsp;.

* [[normal matrix|Normal]] if <math>\boldsymbol{A}^\mathrm{H} \boldsymbol{A} = \boldsymbol{A} \boldsymbol{A}^\mathrm{H}</math>.

* [[Unitary matrix|Unitary]] if <math>\boldsymbol{A}^\mathrm{H} = \boldsymbol{A}^{-1}</math>, equivalently <math>\boldsymbol{A}\boldsymbol{A}^\mathrm{H} = \boldsymbol{I}</math>, equivalently <math>\boldsymbol{A}^\mathrm{H}\boldsymbol{A} = \boldsymbol{I}</math>.

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== Motivation ==

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication:

:<math>a + ib \equiv \begin{pmatrixbmatrix} a & -b \\ b & a \end{pmatrixbmatrix}.</math>

That is, denoting each ''complex'' number ''z'' by the ''real'' 2&times;2 matrix of the linear transformation on the [[Argand diagram]] (viewed as the ''real'' vector space <math>\mathbb{R}^2</math>), affected by complex ''z''-multiplication on <math>\mathbb{C}</math>.

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* <math>(z\boldsymbol{A})^\mathrm{H} = \overline{z} \boldsymbol{A}^\mathrm{H}</math> for any complex number <math>z</math> and any ''m''-by-''n'' matrix <math>\boldsymbol{A}</math>.

* <math>(\boldsymbol{A}\boldsymbol{B})^\mathrm{H} = \boldsymbol{B}^\mathrm{H} \boldsymbol{A}^\mathrm{H}</math> for any ''m''-by-''n'' matrix <math>\boldsymbol{A}</math> and any ''n''-by-''p'' matrix <math>\boldsymbol{B}</math>. Note that the order of the factors is reversed.<ref name=":1" />

* <math>\left(\boldsymbol{A}^\mathrm{H}\right)^\mathrm{H} = \boldsymbol{A}</math> for any ''m''-by-''n'' matrix <math>\boldsymbol{A}</math>, i.e. Hermitian transposition is an [[Involution (mathematics)|involution]].

* If <math>\boldsymbol{A}</math> is a square matrix, then <math>\operatorname{det}\left(\boldsymbol{A}^\mathrm{H}\right) = \overline{\operatorname{det}\left(\boldsymbol{A}\right)}</math> where <math>\operatorname{det}(A)</math> denotes the [[determinant]] of <math>\boldsymbol{A}</math> .

* If <math>\boldsymbol{A}</math> is a square matrix, then <math>\operatorname{tr}\left(\boldsymbol{A}^\mathrm{H}\right) = \overline{\operatorname{tr}(\boldsymbol{A})}</math> where <math>\operatorname{tr}(A)</math> denotes the [[trace (matrix)|trace]] of <math>\boldsymbol{A}</math>.

* <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>\left(\boldsymbol{A}^\mathrm{H}\right)^{-1} = \left(\boldsymbol{A}^{-1}\right)^{\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>\left\langle \boldsymbol{A} x,y \right\rangle_m = \left\langle x, \boldsymbol{A}^\mathrm{H} y\right\rangle_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\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>.

==Generalizations==