Conjugate transpose: Difference between revisions

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{{redirect|Adjoint matrix|the transpose of cofactor|Adjugate matrix}}

In [[mathematics]], the '''conjugate transpose''', (oralso known as the '''Hermitian transpose'''), of an ''<math>m''-by-'' \times n''</math> [[Complex number|complex]] [[matrix (mathematics)|matrix]] <math>\boldsymbol{A}</math> withis [[complexan number|complex]]<math>n entries\times is the ''n''-by-''m''</math> matrix obtained fromby [[transposing]] <math>\boldsymbol{A}</math> by taking the [[transpose]] and then taking theapplying [[complex conjugate]] ofon each entry (the complex conjugate of <math>a+ib</math> being <math>a-ib</math>, for real numbers <math>a</math> and <math>b</math>). It is often denoted as <math>\boldsymbol{A}^\mathrm{H}</math> or <math>\boldsymbol{A}^*</math>.<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Conjugate Transpose|url=https://mathworld.wolfram.com/ConjugateTranspose.html|access-date=2020-09-08|website=mathworld.wolfram.com|language=en}}</ref><ref name=":2">{{Cite web|title=conjugate transpose|url=https://planetmath.org/ConjugateTranspose|access-date=2020-09-08|website=planetmath.org}}</ref>

For [[Real number|real]] matrices, the conjugate transpose is just the transpose, <math>\boldsymbol{A}^\mathrm{H} = \boldsymbol{A}^\mathsf{T}</math>.

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== Motivation ==

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

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

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

Thus, an ''<math>m''-by-'' \times n''</math> matrix of complex numbers could be well represented by a 2''m''-by-2''n''<math>2m \times 2n</math> matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an ''<math>n''-by-'' \times m''</math> matrix made up of complex numbers.

==Properties of the conjugate transpose==

* <math>(\boldsymbol{A} + \boldsymbol{B})^\mathrm{H} = \boldsymbol{A}^\mathrm{H} + \boldsymbol{B}^\mathrm{H}</math> for any two matrices <math>\boldsymbol{A}</math> and <math>\boldsymbol{B}</math> of the same dimensions.

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

* <math>(\boldsymbol{A}\boldsymbol{B})^\mathrm{H} = \boldsymbol{B}^\mathrm{H} \boldsymbol{A}^\mathrm{H}</math> for any ''<math>m''-by-'' \times n''</math> matrix <math>\boldsymbol{A}</math> and any ''<math>n''-by-'' \times p''</math> 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 ''<math>m''-by-'' \times n''</math> 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>\det\left(\boldsymbol{A}^\mathrm{H}\right) = \overline{\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 ''<math>m''-by-'' \times n''</math> 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==