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{{redirect|Adjoint matrix|the transpose of cofactor|Adjugate matrix}}
In [[mathematics]], the '''conjugate transpose''', '''Hermitian transpose''', '''Hermitian conjugate''', '''bedaggered matrix''', or '''adjoint matrix''' of an ''m''-by-''n'' [[matrix (mathematics)|matrix]] ''{{math|A}}'' with [[complex number|complex]] entries is the ''n''-by-''m'' matrix ''{{math|A}}''<sup>*</sup> obtained from ''{{math|A}}'' by taking the [[transpose]] and then taking the [[complex conjugate]] of each entry (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by
:<math>(\mathbf{A}^*)_{ij} = \overline{\mathbf{A}_{ji}}</math>
where the subscripts denote the ''i'',''j''-th entry, for 1 &le; ''i'' &le; ''n'' and 1 &le; ''j'' &le; ''m'', and the overbar denotes a scalar [[complex conjugate]]. (The complex conjugate of <math>a + bi</math>, where ''a'' and ''b'' are reals, is <math>a - bi</math>.)

This definition can also be written as
:<math>\mathbf{A}^* = (\overline{\mathbf{A}})^\mathrm{T} = \overline{\mathbf{A}^\mathrm{T}}</math>
where <math>\mathbf{A}^\mathrm{T} \,\!</math> denotes the transpose and <math>\overline{\mathbf{A}} \,\!</math> denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are ''Hermitian conjugate'', or '''transjugate'''. The conjugate transpose of a matrix ''{{math|A}}'' can be denoted by any of these symbols:
* <math>\mathbf{A}^* \,\!</math> or <math>\mathbf{A}^\mathrm{H} \,\!</math>, commonly used in [[linear algebra]]
* <math>\mathbf{A}^\dagger \,\!</math> (sometimes pronounced as "''{{math|A}}'' [[dagger (typography)|dagger]]"), universally used in [[quantum mechanics]]
* <math>\mathbf{A}^+ \,\!</math>, although this symbol is more commonly used for the [[Moore–Penrose pseudoinverse]]

In some contexts, <math>\mathbf{A}^* \,\!</math> denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by <math>\mathbf{A}^{*\mathrm{T}} \,\!</math> or <math>\mathbf{A}^{\mathrm{T}*} \,\!</math>.

==Example==
If
:<math>\mathbf{A} = \begin{bmatrix} 3 + i & 5 & -2i \\ 2-2i & i & -7-13i \end{bmatrix}</math>
then
:<math>\mathbf{A}^* = \begin{bmatrix} 3-i & 2+2i \\ 5 & -i \\ 2i & -7+13i\end{bmatrix}</math>

==Basic remarks==

A square matrix ''{{math|A}}'' with entries <math>a_{ij}</math> is called
* [[hermitian matrix|Hermitian]] or self-adjoint if ''{{math|A}}'' = ''{{math|A}}''<sup>*</sup>, i.e., <math>a_{ij}=\overline{a_{ji}}</math>&nbsp;.
* [[skew-Hermitian matrix|skew Hermitian]] or antihermitian if ''{{math|A}}'' = &minus;''{{math|A}}''<sup>*</sup>, i.e., <math>a_{ij}=-\overline{a_{ji}}</math>&nbsp;.
* [[normal matrix|normal]] if ''{{math|A}}''<sup>*</sup>''{{math|A}}'' = ''{{math|AA}}''<sup>*</sup>.
* [[Unitary_matrix|unitary]] if ''{{math|A}}''<sup>*</sup> = ''{{math|A}}''<sup>-1</sup>.

Even if ''{{math|A}}'' is not square, the two matrices ''{{math|A}}''<sup>*</sup>''{{math|A}}'' and ''{{math|AA}}''<sup>*</sup> are both Hermitian and in fact [[Positive-definite matrix|positive semi-definite matrices]].

The conjugate transpose "adjoint" matrix ''{{math|A}}''<sup>*</sup> should not be confused with the [[adjugate]] adj(''{{math|A}}''), which is also sometimes called "adjoint".

Finding the conjugate transpose of a matrix ''{{math|A}}'' with [[Real number|real]] entries reduces to finding the [[transpose]] of ''{{math|A}}'', as the conjugate of a real number is the number itself.

== Motivation ==
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2&times;2 real matrices, obeying matrix addition and multiplication:

:<math>a + ib \equiv  \left(\begin{matrix} a & -b \\ b & a \end{matrix}\right). </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>.

An ''m''-by-''n'' matrix of complex numbers could therefore equally well be represented by a ''2m''-by-''2n'' 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 ''n''-by-''m'' matrix made up of complex numbers.

==Properties of the conjugate transpose==
* (''{{math|A}}'' + ''{{math|B}}'')<sup>*</sup> = ''{{math|A}}''<sup>*</sup> + ''{{math|B}}''<sup>*</sup> for any two matrices ''{{math|A}}'' and ''{{math|B}}'' of the same dimensions.
* (''r'' ''{{math|A}}'')<sup>*</sup> =  ''r''<sup>*</sup>''{{math|A}}''<sup>*</sup> for any complex number ''r'' and any matrix ''{{math|A}}''. Here ''r''<sup>*</sup> refers to the complex conjugate of ''r''.
* (''{{math|AB}}'')<sup>*</sup> = ''{{math|B}}''<sup>*</sup>''{{math|A}}''<sup>*</sup> for any ''m''-by-''n'' matrix ''{{math|A}}'' and any ''n''-by-''p'' matrix ''{{math|B}}''. Note that the order of the factors is reversed.
* (''{{math|A}}''<sup>*</sup>)<sup>*</sup> = ''{{math|A}}'' for any matrix ''{{math|A}}''.
* If ''{{math|A}}'' is a square matrix, then [[determinant|det]](''{{math|A}}''<sup>*</sup>) = (det ''{{math|A}}'')<sup>*</sup> and [[trace (matrix)|tr]](''{{math|A}}''<sup>*</sup>) = (tr ''{{math|A}}'')<sup>*</sup>
* ''{{math|A}}'' is [[invertible matrix|invertible]] [[if and only if]] ''{{math|A}}''<sup>*</sup> is invertible, and in that case (''{{math|A}}''<sup>*</sup>)<sup>−1</sup> = (''{{math|A}}''<sup>−1</sup>)<sup>*</sup>.
* The [[eigenvalue]]s of ''{{math|A}}''<sup>*</sup> are the complex conjugates of the [[eigenvalue]]s of ''{{math|A}}''.
* <math>\langle \mathbf{Ax}, \mathbf{y}\rangle = \langle \mathbf{x},\mathbf{A}^* \mathbf{y} \rangle</math> for any ''m''-by-''n'' matrix ''{{math|A}}'', any vector ''x'' in <math> \mathbb{C}^n </math> and any vector ''y'' in <math> \mathbb{C}^m </math>. Here, <math>\langle\cdot,\cdot\rangle</math> denotes the standard complex [[inner product]] on <math> \mathbb{C}^m </math> and <math> \mathbb{C}^n </math>.

==Generalizations==
The last property given above shows that if one views ''{{math|A}}'' as a [[linear transformation]] from Euclidean [[Hilbert space]] <math> \mathbb{C}^n </math> to <math> \mathbb{C}^m </math>, then the matrix ''{{math|A}}''<sup>*</sup> corresponds to the [[Hermitian adjoint|adjoint operator]] of ''{{math|A}}''. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices.

Another generalization is available: suppose ''{{math|A}}'' is a linear map from a complex [[vector space]] ''{{math|V}}'' to another, ''{{math|W}}'', then the [[complex conjugate linear map]] as well as the [[transpose of a linear map|transposed linear map]] are defined, and we may thus take the conjugate transpose of ''{{math|A}}'' to be the complex conjugate of the transpose of ''{{math|A}}''. It maps the conjugate [[dual space|dual]] of ''{{math|W}}'' to the conjugate dual of ''{{math|V}}''.

==See also==
*[[Hermitian adjoint|Hermitian conjugate]]
*[[Adjugate matrix]]

==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]]
[[Category:Matrices]]

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