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{{short description|Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry}}

In [[mathematics]], the '''conjugate transpose''' or '''adjoint''' of an ''m''-by-''n'' [[matrix (mathematics)|matrix]] ''A'' with [[complex number|complex]] entries is the ''n''-by-''m'' matrix ''A''^* obtained from ''A'' by taking the [[transpose]] and then taking the [[complex conjugate]] of each entry. Formally

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

:<math>(A^H)[i,j] = \overline{A[j,i]}</math>

for 1 &le; ''i'' &le; ''n'' and 1 &le; ''j'' &le; ''m''.

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>\mathbf{A}</math> is an <math>n \times m</math> matrix obtained by [[transpose|transposing]] <math>\mathbf{A}</math> and applying [[complex conjugate|complex conjugation]] to 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>). There are several notations, such as <math>\mathbf{A}^\mathrm{H}</math> or <math>\mathbf{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> <math>\mathbf{A}'</math>,<ref>

This definition can also be written as

H. W. Turnbull, A. C. Aitken,

:<math> A^H \equiv {\overline A}^{T}</math>

"An Introduction to the Theory of Canonical Matrices,"

where <math>A^T \,\!</math> denotes the transpose and <math> \overline A \,\!</math> denotes the matrix with complex conjugated entries.

1932.

</ref> or (often in physics) <math>\mathbf{A}^{\dagger}</math>.

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

Alternative names for the conjugate transpose of a matrix are '''adjoint matrix''', '''Hermitian conjugate''', or '''tranjugate'''. The conjugate transpose of a matrix ''A'' can be denoted by any of these symbols:

* <math>A^H \,\!</math>, commonly used in [[linear algebra]]

* <math>A^\dagger \,\!</math>, universally used in [[quantum mechanics]]

* <math>A^* \,\!</math> (which can also denote the complex conjugate, however)

==ExampleDefinition==

The conjugate transpose of an <math>m \times n</math> matrix <math>\mathbf{A}</math> is formally defined by

{{Equation box 1

|indent =

|title=

|equation = {{NumBlk||<math>\left(\mathbf{A}^\mathrm{H}\right)_{ij} = \overline{\mathbf{A}_{ji}}</math>|{{EquationRef|Eq.1}}}}

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

where the subscript <math>ij</math> denotes the <math>(i,j)</math>-th entry (matrix element), for <math>1 \le i \le n</math> and <math>1 \le j \le m</math>, and the overbar denotes a scalar complex conjugate.

If

:<math>A=\begin{bmatrix}3+i&2\\

2-2i&i\end{bmatrix}</math>

then

:<math>A^H=\begin{bmatrix}3-i&2+2i\\

2&-i\end{bmatrix}.</math>

This definition can also be written as

==Basic remarks==

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

where <math>\mathbf{A}^\operatorname{T}</math> denotes the transpose and <math>\overline{\mathbf{A}}</math> denotes the matrix with complex conjugated entries.

If the entries of ''A'' are real, then ''A''^* coincides with the transpose ''A''^T of ''A''. It is often useful to think of [[Matrix_(mathematics)#Square_matrices_and_related_definitions|square]] complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of [[complex conjugate|complex conjugation]].

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

A square matrix ''A'' is called

* <math>\mathbf{A}^*</math>, commonly used in [[linear algebra]]

* [[hermitian matrix|Hermitian]] or self-adjoint if ''A'' = ''A''^*;

* <math>\mathbf{A}^\mathrm{H}</math>, commonly used in linear algebra

* [[skew-Hermitian matrix|skew Hermitian]] if ''A'' = -''A''^*;

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

* [[normal matrix|normal]] if ''A^*A'' = ''AA^*''.

* <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 only complex conjugated entries and no transposition.

Even if ''A'' is not square, the two matrices ''A^*A'' and ''AA^*'' are both Hermitian and in fact [[positive semi-definite]].

==Example==

The adjoint matrix ''A''^* should not be confused with the [[adjugate]] adj(''A'') (which in older texts is also sometimes called "adjoint").

Suppose we want to calculate the conjugate transpose of the following matrix <math>\mathbf{A}</math>.

:<math>\mathbf{A} = \begin{bmatrix} 1 & -2 - i & 5 \\ 1 + i & i & 4-2i \end{bmatrix}</math>

We first transpose the matrix:

:<math>\mathbf{A}^\operatorname{T} = \begin{bmatrix} 1 & 1 + i \\ -2 - i & i \\ 5 & 4-2i\end{bmatrix}</math>

Then we conjugate every entry of the matrix:

:<math>\mathbf{A}^\mathrm{H} = \begin{bmatrix} 1 & 1 - i \\ -2 + i & -i \\ 5 & 4+2i\end{bmatrix}</math>

==Basic Motivation remarks==

A square matrix <math>\mathbf{A}</math> with entries <math>a_{ij}</math> is called

* [[hermitian matrix|Hermitian]] or [[self-adjoint_operator|self-adjoint]] if <math>\mathbf{A}=\mathbf{A}^\mathrm{H}</math>; i.e., <math>a_{ij} = \overline{a_{ji}}</math>.

* [[skew-Hermitian matrix|Skew Hermitian]] or antihermitian if <math>\mathbf{A}=-\mathbf{A}^\mathrm{H}</math>; i.e., <math>a_{ij} = -\overline{a_{ji}}</math>.

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

* [[Unitary matrix|Unitary]] if <math>\mathbf{A}^\mathrm{H} = \mathbf{A}^{-1}</math>, equivalently <math>\mathbf{A}\mathbf{A}^\mathrm{H} = \boldsymbol{I}</math>, equivalently <math>\mathbf{A}^\mathrm{H}\mathbf{A} = \boldsymbol{I}</math>.

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

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

The conjugate transpose "adjoint" matrix <math>\mathbf{A}^\mathrm{H}</math> should not be confused with the [[adjugate]], <math>\operatorname{adj}(\mathbf{A})</math>, which is also sometimes called ''adjoint''.

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

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

An ''m''-by-''n'' matrix of complex numbers could therefore equally well be represented by a ''2m''-by-''2n'' matrix of real numbers. It therefore arises very naturally that when transposing such a matrix which is made up of complex numbers, one may in the process also have to take the complex conjugate of each entry.

<math display="block">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>.

==Properties of the conjugate transpose==

Thus, an <math>m \times n</math> matrix of complex numbers could be well represented by a <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 \times m</math> matrix made up of complex numbers.

* (''A'' + ''B'')^* = ''A''^* + ''B''^* for any two matrices ''A'' and ''B'' of the same format.

* (''rA'')^* = ''r''^*''A''^* for any complex number ''r'' and any matrix ''A''. Here ''r''^* refers to the complex conjugate of ''r''.

* (''AB'')^* = ''B''^*''A''^* for any ''m''-by-''n'' matrix ''A'' and any ''n''-by-''p'' matrix ''B''. Note that the order of the factors is reversed.

* (''A''^*)^* = ''A'' for any matrix ''A''.

* If ''A'' is a square matrix, then [[determinant|det]] (''A''^*) = (det A)^* and [[trace (matrix)|trace]] (''A''^*) = (trace A)^*

* ''A'' is [[invertible matrix|invertible]] if and only if ''A''^* is invertible, and in that case we have (''A''^*)^-1 = (''A''^-1)^*.

* The [[eigenvalue]]s of ''A''^* are the complex conjugates of the eigenvalues of ''A''.

* <''Ax'',''y''> = <''x'', ''A''^*''y''> for any ''m''-by-''n'' matrix ''A'', any vector ''x'' in '''C'''^''n'' and any vector ''y'' in '''C'''^''m''. Here <.,.> denotes the ordinary Euclidean [[inner product]] (or dot product) on '''C'''^''m'' and '''C'''^''n''.

For an explanation of the notation used here, we begin by representing complex numbers <math>e^{i\theta}</math> as the [[rotation matrix]], that is,

==Generalizations==

<math display="block">

e^{i\theta} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} = \cos \theta \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \sin \theta \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

</math>

Since <math>e^{i\theta} = \cos \theta + i \sin \theta</math>, we are led to the matrix representations of the unit numbers as

<math display="block">

1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad i = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

</math>

A general complex number <math>z=x+iy</math> is then represented as <math>

The last property given above shows that if one views ''A'' as a [[linear map]] from the Euclidean [[Hilbert space]] '''C'''^''n'' to '''C'''^''m'', then the matrix ''A''^* corresponds to the [[Hermitian adjoint|adjoint operator]] of ''A''. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices.

z = \begin{pmatrix} x & -y \\ y & x \end{pmatrix}.

</math> The [[complex conjugate]] operation (that sends <math>a + bi</math> to <math>a - bi</math> for real <math>a, b</math>) is encoded as the matrix transpose.<ref>{{cite book |last=Chasnov |first=Jeffrey R. |url=https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/02%3A_II._Linear_Algebra/01%3A_Matrices/1.06%3A_Matrix_Representation_of_Complex_Numbers |title=Applied Linear Algebra and Differential Equations |date=4 February 2022 |publisher=LibreTexts |contribution=1.6: Matrix Representation of Complex Numbers}}</ref>

==Properties==

Another generalization is available: suppose ''A'' is a linear map from a complex [[vector space]] ''V'' to another ''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 ''A'' to be the complex conjugate of the transpose of ''A''. It maps the conjugate [[dual space|dual]] of ''W'' to the conjugate dual of ''V''.

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

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

* <math>(\mathbf{A}\boldsymbol{B})^\mathrm{H} = \boldsymbol{B}^\mathrm{H} \mathbf{A}^\mathrm{H}</math> for any <math>m \times n</math> matrix <math>\mathbf{A}</math> and any <math>n \times p</math> matrix <math>\boldsymbol{B}</math>. Note that the order of the factors is reversed.<ref name=":1" />

* <math>\left(\mathbf{A}^\mathrm{H}\right)^\mathrm{H} = \mathbf{A}</math> for any <math>m \times n</math> matrix <math>\mathbf{A}</math>, i.e. Hermitian transposition is an [[Involution (mathematics)|involution]].

* If <math>\mathbf{A}</math> is a square matrix, then <math>\det\left(\mathbf{A}^\mathrm{H}\right) = \overline{\det\left(\mathbf{A}\right)}</math> where <math>\operatorname{det}(A)</math> denotes the [[determinant]] of <math>\mathbf{A}</math> .

* If <math>\mathbf{A}</math> is a square matrix, then <math>\operatorname{tr}\left(\mathbf{A}^\mathrm{H}\right) = \overline{\operatorname{tr}(\mathbf{A})}</math> where <math>\operatorname{tr}(A)</math> denotes the [[trace (matrix)|trace]] of <math>\mathbf{A}</math>.

* <math>\mathbf{A}</math> is [[invertible matrix|invertible]] [[if and only if]] <math>\mathbf{A}^\mathrm{H}</math> is invertible, and in that case <math>\left(\mathbf{A}^\mathrm{H}\right)^{-1} = \left(\mathbf{A}^{-1}\right)^{\mathrm{H}}</math>.

* The [[eigenvalue]]s of <math>\mathbf{A}^\mathrm{H}</math> are the complex conjugates of the [[eigenvalue]]s of <math>\mathbf{A}</math>.

* <math>\left\langle \mathbf{A} x,y \right\rangle_m = \left\langle x, \mathbf{A}^\mathrm{H} y\right\rangle_n </math> for any <math>m \times n</math> matrix <math>\mathbf{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==

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

Another generalization is available: suppose <math>A</math> is a linear map from a complex [[vector space]] <math>V</math> to another, <math>W</math>, 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</math> to be the complex conjugate of the transpose of <math>A</math>. It maps the conjugate [[dual space|dual]] of <math>W</math> to the conjugate dual of <math>V</math>.

==See also==

*[[Complex dot product]]

*[[Hermitian adjoint|Hermitian conjugate]]

*[[Hermitian adjoint]]

*[[Adjugate matrix]]

== References ==

<references />

==External links==

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

* [http://mathworld.wolfram.com/ConjugateTranspose.html Conjugate transpose] on Mathworld, Wolfram research

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

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