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→Definition: Added a fact that an entry is a matrix element, and explicitly mentioned one important name of conjugate transpose in a middle of this Wikipedia page even if it is mentioned in the introductory section once. |
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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:<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>
:<math>a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.</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.
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</math>
Since <math>
e^{i\theta} = \cos \theta + i \sin \theta</math>, we are led to the matrix representations of the unit numbers as▼
▲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▼
▲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.▼
▲The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.
==Properties==
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