Conjugate transpose: Difference between revisions

Revision as of 03:20, 19 January 2023

Revision as of 18:51, 18 March 2023

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

In [[mathematics]], the '''conjugate transpose''', also known as the '''Hermitian transpose''', of an <math>m \times n</math> [[Complex number|complex]] [[matrix (mathematics)|matrix]] <math>\boldsymbol{A}</math> is an <math>n \times m</math> matrix obtained by [[transpose|transposing]] <math>\boldsymbol{A}</math> and applying [[complex conjugate]] on 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>

or

<math>\boldsymbol{A}'</math>,<ref>

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where the subscript <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" />

:<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''', '''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

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