Content deleted Content added
m Rev to say "several notations." |
|||
Line 64:
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.
Complex numbers can be usefully represented by \(2 \times 2\) real matrices, obeying matrix addition and multiplication:
\[a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.\]
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 \(\mathbb{C}\).
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 \(e^{i\theta}\) 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.
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==
| |||