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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 obtained from <math>\boldsymbol{A}</math> by taking the [[transpose]] and then taking the [[complex conjugate]] of 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>.
For real matrices, the conjugate transpose is just the transpose, <math>\boldsymbol{A}^\mathrm{H} = \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
* <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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