Conjugate transpose: Difference between revisions

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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>\boldsymbolmathbf{A}</math> is an <math>n \times m</math> matrix obtained by [[transpose|transposing]] <math>\boldsymbolmathbf{A}</math> and applying [[complex conjugate|complex conjugation]] onto 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>). ItThere isare oftenseveral notations, denotedsuch as <math>\boldsymbolmathbf{A}^\mathrm{H}</math> or <math>\boldsymbolmathbf{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>

or

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

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

"An Introduction to the Theory of Canonical Matrices,"

1932.

</ref> andor very commonly(often in physics as) <math>\boldsymbolmathbf{A}^{\dagger}</math>.

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

==Definition==

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

{{Equation box 1

|indent =

|title=

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

|cellpadding= 6

|border

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|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.

This definition can also be written as

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

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

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>\boldsymbolmathbf{A}</math> can be denoted by any of these symbols:

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

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

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

* <math>\boldsymbolmathbf{A}^+</math>, although this symbol is more commonly used for the [[Moore–Penrose pseudoinverse]]

In some contexts, <math>\boldsymbolmathbf{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>\boldsymbolmathbf{A}</math>.

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

We first transpose the matrix:

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

Then we conjugate every entry of the matrix:

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

==Basic remarks==

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

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

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

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

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

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

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

The conjugate transpose ofcan abe matrixmotivated <math>\boldsymbol{A}</math>by withnoting [[realthat number|real]]complex entriesnumbers reducescan tobe theusefully [[transpose]]represented ofby <math>2 \boldsymbol{A}times 2</math>, asreal thematrices, conjugateobeying of[[matrix a real number is theaddition]] numberand itself.multiplication:

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

== Motivation ==

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:

:<math>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>.

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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.

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,

==Properties of the conjugate transpose==

<math display="block">

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

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

</math>

* <math>(\boldsymbol{A}\boldsymbol{B})^\mathrm{H} = \boldsymbol{B}^\mathrm{H} \boldsymbol{A}^\mathrm{H}</math> for any <math>m \times n</math> matrix <math>\boldsymbol{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" />

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

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

<math display="block">

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

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

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

</math>

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

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

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

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

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==

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

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

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

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

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

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

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

* <math>\left\langle \boldsymbolmathbf{A} x,y \right\rangle_m = \left\langle x, \boldsymbolmathbf{A}^\mathrm{H} y\right\rangle_n </math> for any <math>m \times n</math> matrix <math>\boldsymbolmathbf{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>\boldsymbolmathbf{A}</math> as a [[linear transformation]] from [[Hilbert space]] <math> \mathbb{C}^n </math> to <math> \mathbb{C}^m ,</math> then the matrix <math>\boldsymbolmathbf{A}^\mathrm{H}</math> corresponds to the [[Hermitian adjoint|adjoint operator]] of <math>\boldsymbolmathbf 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>.

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