Conjugate transpose: Difference between revisions

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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

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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

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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>

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,

<math display="block">

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,

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>

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">

Since

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

<math>

e^{i\theta} =\cos \theta+\sin \theta</math>

we are led to the matrix representations of the unit numbers as

<math>

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>

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

</math>

The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.

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

<ref>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>

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 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>.

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[[Category:Linear algebra]]

[[Category:Matrices (mathematics)]]