Zeriegen Sie die folgenden Matrizen in ihre Real- und Imaginärteile und prüfen Sie dann, welche Matrizen hermitesch bzw. schiefhermitesch sind: \(\mathbf{A}=\left(\begin{array}{cc}-\mathrm{j} & -5+2 \mathrm{j} \\ 5+2 \mathrm{j} & -4 \mathrm{j}\end{array}\right), \quad \mathbf{B}=\left(\begin{array}{cc}1 & 2-4 \mathrm{j} \\ 2+4 \mathrm{j} & 2\end{array}\right)\) \(C=\left(\begin{array}{rrr}-j & 0 & 0 \\ 0 & -j & 0 \\ 0 & 0 & -j\end{array}\right), \quad D=\left(\begin{array}{ccc}1 & -j & 1-2 j \\ j & 2 & 4 \\ 1+2 j & 4 & 3\end{array}\right)\) Welchen Wert besitzen die zugehôrigen Determinanten?

Let the given matrices be as follows:\[ \mathbf{A}=\left(\begin{array}{cc}-\mathrm{j} & -5+2 \mathrm{j} \ 5+2 \mathrm{j} & -4 \mathrm{j}\end{array}\right), \quad \mathbf{B}=\left(\begin{array}{cc}1 & 2-4 \mathrm{j} \ 2+4 \mathrm{j} & 2\end{array}\right) \] \[ \mathbf{C}=\left(\begin{array}{rrr} -j & 0 & 0 \ 0 & -j & 0 \ 0 & 0 & -j \end{array}\right), \quad \mathbf{D}=\left(\begin{array}{ccc} 1 & -j & 1-2 j \ j & 2 & 4 \ 1+2 j & 4 & 3 \end{array}\right) \]

The real part of a matrix is obtained by taking the real part of each element individually. For matrix \( \mathbf{A} \): \[ \text{Re}(\mathbf{A})=\left(\begin{array}{cc} 0 & -5 \ 5 & 0 \end{array}\right) \] For matrix \( \mathbf{B} \): \[ \text{Re}(\mathbf{B})=\left(\begin{array}{cc} 1 & 2 \ 2 & 2 \end{array}\right) \] For matrix \( \mathbf{C} \): \[ \text{Re}(\mathbf{C})=\left(\begin{array}{rrr} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{array}\right) \] For matrix \( \mathbf{D} \): \[ \text{Re}(\mathbf{D})=\left(\begin{array}{ccc} 1 & 0 & 1 \ 0 & 2 & 4 \ 1 & 4 & 3 \end{array}\right) \]

The imaginary part of a matrix is obtained by taking the coefficient of \( \mathrm{j} \) in each element. For matrix \( \mathbf{A} \): \[ \text{Im}(\mathbf{A})=\left(\begin{array}{cc} -1 & 2 \ 2 & -4 \end{array}\right) \] For matrix \( \mathbf{B} \): \[ \text{Im}(\mathbf{B})=\left(\begin{array}{cc} 0 & -4 \ 4 & 0 \end{array}\right) \] For matrix \( \mathbf{C} \): \[ \text{Im}(\mathbf{C})=\left(\begin{array}{rrr} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{array}\right) \] For matrix \( \mathbf{D} \): \[ \text{Im}(\mathbf{D})=\left(\begin{array}{ccc} 0 & -1 & -2 \ 1 & 0 & 0 \ 2 & 0 & 0 \end{array}\right) \]

A matrix \( \mathbf{M} \) is Hermitian if \( \mathbf{M} = \mathbf{M}^{\dagger} \) and skew-Hermitian if \( \mathbf{M} = -\mathbf{M}^{\dagger} \), where \( \mathbf{M}^{\dagger} \) is the conjugate transpose of \( \mathbf{M} \). For matrix \( \mathbf{A} \), \( \mathbf{A}^{\dagger}eq \mathbf{A} \) and \( \mathbf{A}^{\dagger}=-\mathbf{A} \), so \( \mathbf{A} \) is skew-Hermitian. For matrix \( \mathbf{B} \), neither \( \mathbf{B}^{\dagger}=\mathbf{B} \) nor \( \mathbf{B}^{\dagger}=-\mathbf{B} \), so \( \mathbf{B} \) is neither Hermitian nor skew-Hermitian. For matrix \( \mathbf{C} \), \( \mathbf{C}^{\dagger}=\mathbf{C} \), so \( \mathbf{C} \) is Hermitian and \( \mathbf{C} \) is skew-Hermitian since \( \mathbf{C}^{\dagger}=-\mathbf{C} \). For matrix \( \mathbf{D} \), neither \( \mathbf{D}^{\dagger}=\mathbf{D} \) nor \( \mathbf{D}^{\dagger}=-\mathbf{D} \). so \( \mathbf{D} \) is neither Hermitian nor skew-Hermitian.

Determine the determinants of the matrices: For matrix \( \mathbf{A} \):\[ \text{det}(\mathbf{A})=(-\mathrm{j})(-4\mathrm{j}) - (-5+2\mathrm{j})(5+2\mathrm{j}) = -12 \] For matrix \( \mathbf{B} \):\[ \text{det}(\mathbf{B})=1(2) - (2-4\mathrm{j})(2+4\mathrm{j}) = -15 \] For matrix \( \mathbf{C} \):\[ \text{det}(\mathbf{C})=(-\mathrm{j})^3 = -\mathrm{j}^3 = 1 \] For matrix \( \mathbf{D} \):\[ \text{det}(\mathbf{D}) = 1* (2*3 - 4*4) - (-\mathrm{j}(24 - 4) + (1-2\mathrm{j})*4\mathrm{j} -50-10\mathrm{j} - 4 ) = 0 \]