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The conjugate transpose of a matrix <math>\mathbf{A}</math> with [[real number|real]] entries reduces to the [[transpose]] of <math>\mathbf{A}</math>, as the conjugate of a real number is the number itself.
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:
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>.
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">
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▼
▲e^{i\theta} = \cos \theta + i \sin \theta</math>, 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}.
</math>
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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 (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==
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