Conjugate transpose: Difference between revisions

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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.

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>

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

</math>

Since

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

<math>

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

Since \(e^{i\theta} = \cos \theta + i \sin \theta\), 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>

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

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

</math>

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

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>.

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

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

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>.

==Properties of the conjugate transpose==