Conjugate transpose: Difference between revisions

Revision as of 16:25, 23 February 2024

Latest revision as of 02:17, 10 August 2025

(16 intermediate revisions by 12 users not shown)

Line 21:

|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

Line 28:

where <math>\mathbf{A}^\operatorname{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 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:

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

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

Line 55:

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''.

The conjugate transpose ofcan abe matrixmotivated <math>\mathbf{A}</math>by withnoting [[realthat number|real]]complex entriesnumbers reducescan tobe theusefully [[transpose]]represented ofby <math>2 \mathbf{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>.

Line 65 ⟶ 62:

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,

Complex numbers can be usefully represented by \(2 \times 2\) real matrices, obeying matrix addition and multiplication:

<math display="block">

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

\[a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.\]

</math>

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

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 \(\mathbb{R}^2\)), affected by complex \(z\)-multiplication on \(\mathbb{C}\).

<math display="block">

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

Thus, an \(m \times n\) matrix of complex numbers could be well represented by a \(2m \times 2n\) 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 an \(n \times m\) matrix made up of complex numbers.

</math>

For an explanation of the notation used here, we begin by representing complex numbers \(e^{i\theta}\) as the rotation matrix, that is,

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

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

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

A general complex number \(z = x + iy\) is then represented as

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

The complex conjugate operation, where \(i \rightarrow -i\), is seen to be just the matrix transpose.

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

For further reference, see <ref>[here](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)</ref>.

\[a +z ib \equiv= \begin{bmatrixpmatrix} ax & -by \\ by & ax \end{bmatrixpmatrix}.\]

</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 of the conjugate transpose==

* <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>.

Line 116 ⟶ 102:

[[Category:Linear algebra]]

[[Category:Matrices (mathematics)]]