Conjugate transpose: Difference between revisions

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

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

This is a particular case of the '''[[Hermitian conjugate]]''' (sometimes called [[Hermitian adjoint]] or just '''adjoint''') [[linear operator]].

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>

More generally, if we have a [[linear map]] A from a complex vector space V to another W, the conjugate transpose of A is the [[complex conjugate linear map|conjugate]] of the [[transpose of a linear map|transpose]] of A. It maps the conjugate dual of W to the conjugate dual of V.

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

"An Introduction to the Theory of Canonical Matrices,"

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>.

==Example==

==Definition==

For example, if

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

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

{{Equation box 1

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

|indent =

then

|title=

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

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

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

|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.

==Basic remarks==

This definition can also be written as

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

:<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.

The square matrix ''A'' is called [[hermitian matrix|hermitian]] or self-adjoint if ''A'' = ''A''^*. It is called [[normal matrix|normal]] if ''A^*A'' = ''AA^*''.

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:

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

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

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

* <math>\mathbf{A}^\dagger</math> (sometimes pronounced as ''A [[dagger (typography)|dagger]]''), commonly 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 only complex conjugated entries and no transposition.

==Example==

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

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''.

==Properties of the conjugate transpose==

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:

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

<math display="block">a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.</math>

* (''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''.

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

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>.

==Adjoint operator in Hilbert space==

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.

The final property given above shows that if one views ''A'' as a [[linear operator]] from the Euclidean [[Hilbert space]] '''C'''^''n'' to '''C'''^''m'', then the matrix ''A''^* corresponds to the '''adjoint operator'''.

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,

For an operator ''A'' on a [[Hilbert space]] ''H'', the relation

<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>

: <math> \langle A x, y \rangle = \langle x, A^* y \rangle </math>

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==

can be used to define the ''adjoint'' ''A''^*, by means of the [[Riesz representation theorem]]. This definition can be extended even for operators which are not bounded. See [[self-adjoint operator]] for the details.

* <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 notation <math>A^{\dagger}</math> is also used to denote the adjoint of ''A'', especially when used in conjunction with the [[bra-ket notation]]. The adjoint condition takes the form:

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>.

:<math> \lang \phi | (A|\psi) \rang = \lang (A^{\dagger}\phi) | \psi) \rang. </math>

==See also==

The term '''Hermitian conjugate transpose''' is also sometimes used to refer to the adjoint. Although the etymology of this usage is not clear, it has been suggested that it results from the expression [[Hermitian]] operator being used to denote self-adjoint operators, that is operators ''A'' for which

*[[Complex dot product]]

*[[Hermitian adjoint]]

*[[Adjugate matrix]]

== References ==

:<math> A = A^{\dagger} </math>.

<references />

==External links==

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

Note that there is a general theory of [[adjoint functor]]s in [[category theory]] which includes the previous definition as a special case. See [[John Baez]]' expository article [http://math.ucr.edu/home/baez/week78.html week78] for a discussion of this, and earlier writings for introductory material on category theory.

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