Conjugate transpose: Difference between revisions

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

In [[mathematics]], the '''conjugate transpose''' (or '''Hermitian transpose''') of an ''m''-by-''n'' [[matrix (mathematics)|matrix]] <math>\boldsymbol{A}</math> with [[complex number|complex]] entries, is the ''n''-by-''m'' matrix <math>\boldsymbol{A}^\mathrm{H}</math> obtained from <math>\boldsymbol{A}</math> by taking the [[transpose]] and then taking the [[complex conjugate]] of each entry. (Thethe complex conjugate of <math>a+ib</math> isbeing <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=":0">{{Cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-09-08|website=Math Vault|language=en-US}}</ref><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 matrices, the conjugate transpose is just the transpose, <math>\boldsymbol{A}^\mathrm{H} = \boldsymbol{A}^\mathsf{T}</math>.

==Definition==

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where the subscripts denote 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" />

:<math>\boldsymbol{A}^\mathrm{H} = \left(\overline{\boldsymbol{A}}\right)^\mathsf{T} = \overline{\boldsymbol{A}^\mathsf{T}}</math>

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Other names for the conjugate transpose of a matrix are '''Hermitian conjugate''', '''bedaggered matrix''', '''adjoint matrix''' or '''transjugate'''. The conjugate transpose of a matrix <math>\boldsymbol{A}</math> can be denoted by any of these symbols:

* <math>\boldsymbol{A}^*</math>, commonly used in [[linear algebra]]<ref name=":2" />

* <math>\boldsymbol{A}^\mathrm{H}</math>, commonly used in linear algebra<ref name=":0" />

* <math>\boldsymbol{A}^\dagger</math> (sometimes pronounced as ''A [[dagger (typography)|dagger]]''), commonly used in [[quantum mechanics]]

* <math>\boldsymbol{A}^+</math>, although this symbol is more commonly used for the [[Moore–Penrose pseudoinverse]]

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

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

* [[Unitary matrix|unitaryUnitary]] 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>.

Even if <math>\boldsymbol{A}</math> is not square, the two matrices <math>\boldsymbol{A}^\mathrm{H}\boldsymbol{A}</math> and <math>\boldsymbol{A}\boldsymbol{A}^\mathrm{H}</math> are both Hermitian and in fact [[positive-definite matrix|positive semi-definite matrices]].

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

AnTHus, an ''m''-by-''n'' matrix of complex numbers could therefore equallybe well be represented by a 2''m''-by-2''n'' matrix of real numbers. The conjugate transpose therefore arises very naturally as the result of simply transposing such a matrix, whenmatrix—when viewed back again as ''n''-by-''m'' 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 ''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>(\boldsymbol{A}^\mathrm{H})^\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}(\boldsymbol{A}^\mathrm{H}) = \overline{\operatorname{det}(\boldsymbol{A})}</math> where <math>\operatorname{det}(A)</math> denotes the [[determinant]] of <math>\boldsymbol{A}</math> .

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==External links==

* {{springer|title=Adjoint matrix|id=p/a010850}}

* {{MathWorld | urlname=ConjugateTranspose | title=Conjugate Transpose}}

* {{planetmath reference|id=4382|title=Conjugate transpose}}

[[Category:Linear algebra]]