Bilden Sie die jeweils 7ugeh?rige konjugiert komplexe Matrix \(\mathbf{A}^{*}\) sowie die konjugiert transponierte Matrix \(\bar{A}\) : a) \(\quad \mathbf{A}=\left(\begin{array}{ccc}2+j & 1+2 j & j \\ 0 & 2-2 j & 1 \\\ 1-j & 5+3 j & j\end{array}\right)\) b) \(\quad \mathbf{A}=\left(\begin{array}{rl}10+5 j & 1-2 j \\ 3-2 j & 4+2 j \\\ 5+5 j & 1-2 j\end{array}\right)\)

Understanding the concept of complex conjugate is crucial when dealing with complex matrices. A complex number is written in the form of \(a + bj\), where \(a\) is the real part and \(b\) is the imaginary part.
The complex conjugate of \(a + bj\) is \(a - bj\), which simply changes the sign of the imaginary part while keeping the real part unchanged.
This operation is important because it reveals symmetries and properties of complex numbers and matrices. For instance, it is used to find the magnitude of a complex number or to work with Hermitian matrices.

A matrix transpose is another fundamental operation in linear algebra.
Transposing a matrix means swapping its rows and columns.
In other words, the \( (i, j) \) element of the matrix becomes the \( (j, i) \) element in the transposed matrix.
For matrix \(\textbf{A}\) represented as:
\[\textbf{A}=\begin{pmatrix}a & b & c \ d & e & f \ g & h & i\end{pmatrix}\]
The transpose of \(\textbf{A}\), denoted as \( \textbf{A}^T \) or \( \textbf{A}^{T} \), is:
\[\textbf{A}^T=\begin{pmatrix}a & d & g \ b & e & h \ c & f & i\end{pmatrix}\]
This operation is useful in numerous mathematical and engineering applications, including solving systems of linear equations and working with different forms of matrix decompositions. Remember this is just swapping rows with columns.

The conjugate transpose (also known as Hermitian transpose) combines both the conjugate and transpose operations.
For a given matrix \( \textbf{A} \), we first take the transpose, and then compute the complex conjugate of each element.
Let's see step by step how it is done.
Given matrix \( \textbf{A} = \begin{pmatrix}2+j & 1+2j & j \ 0 & 2-2j & 1 \ 1-j & 5+3j & jd{pmatrix} \)
First, we take the transpose of \( \textbf{A} \):
\[ \textbf{A}^{T}= \begin{pmatrix}2+j & 0 & 1-j \ 1+2j & 2-2j & 5+3j \ j & 1 & jd{pmatrix} \]
Then, we compute the complex conjugate of each element to get the conjugate transpose \( \bar{A} \):
\[ \bar{A}= \begin{pmatrix}2-j & 0 & 1+j \ 1-2j & 2+2j & 5-3j \ -j & 1 & -jd{pmatrix} \]
The conjugate transpose is important in various fields, especially in quantum mechanics and signal processing.
It helps in defining inner product spaces and understanding properties like unitary and Hermitian matrices.