Q1: Find the Conjugate of Matrix A, where:
A = [(2 + 3i, 4 – i), (5 + 2i, 1 – 4i)]
Q2: Find the Conjugate of Matrix B, where:
B = [(3 + 4i, 2 – 3i), (7 – i, 6 + 2i)]
Q3: Determine if Matrix A is Hermitian, where:
A = [(3 + 2i, 1 – 4i), (1 + 4i, 5 – 3i)]
Q4: Determine if Matrix B is Hermitian, where:
B = [(4 + i, 2 – 3i), (2 + 3i, 7 – 2i)]
Q5: Find the Hermitian of Matrix C, where:
C = [(1 + 2i, 3 – i), (3 + i, 2 – 4i)]
Q6: Find the Conjugate and Hermitian of Matrix D, where:
D = [(1, 2 + i), (2 – i, 3)]
Q7: Check if Matrix E is Hermitian, where:
E = [(5 + i, 2), (2, 5 – i)]
Solution:
\[\] \[
\textbf{Exercise: Conjugate and Hermitian Matrices}
\]
\[
\textbf{Q1: Find the Conjugate of Matrix A, where:}
\]
\[
A = \begin{pmatrix} 2 + 3i & 4 – i \\ 5 + 2i & 1 – 4i \end{pmatrix}
\]
\[
\textbf{Solution:}
\]
To find the conjugate of a matrix, take the conjugate of each element.
\[
\overline{A} = \begin{pmatrix} \overline{2 + 3i} & \overline{4 – i} \\ \overline{5 + 2i} & \overline{1 – 4i} \end{pmatrix}
= \begin{pmatrix} 2 – 3i & 4 + i \\ 5 – 2i & 1 + 4i \end{pmatrix}
\]
\[
\textbf{Q2: Find the Conjugate of Matrix B, where:}
\]
\[
B = \begin{pmatrix} 3 + 4i & 2 – 3i \\ 7 – i & 6 + 2i \end{pmatrix}
\]
\[
\textbf{Solution:}
\]
\[
\overline{B} = \begin{pmatrix} 3 – 4i & 2 + 3i \\ 7 + i & 6 – 2i \end{pmatrix}
\]
\[
\textbf{Q3: Determine if Matrix A is Hermitian, where:}
\]
\[
A = \begin{pmatrix} 3 + 2i & 1 – 4i \\ 1 + 4i & 5 – 3i \end{pmatrix}
\]
\[
\textbf{Solution:}
\]
To check if the matrix is Hermitian, we compare the matrix to its conjugate transpose.
\[
A^H = \overline{A^T} = \begin{pmatrix} 3 – 2i & 1 + 4i \\ 1 – 4i & 5 + 3i \end{pmatrix}
\]
Since \( A^H = A \), matrix A is Hermitian.
\[
\textbf{Q4: Determine if Matrix B is Hermitian, where:}
\]
\[
B = \begin{pmatrix} 4 + i & 2 – 3i \\ 2 + 3i & 7 – 2i \end{pmatrix}
\]
\[
\textbf{Solution:}
\]
\[
B^H = \overline{B^T} = \begin{pmatrix} 4 – i & 2 + 3i \\ 2 – 3i & 7 + 2i \end{pmatrix}
\]
Since \( B^H \neq B \), matrix B is not Hermitian.
\[
\textbf{Q5: Find the Hermitian of Matrix C, where:}
\]
\[
C = \begin{pmatrix} 1 + 2i & 3 – i \\ 3 + i & 2 – 4i \end{pmatrix}
\]
\[
\textbf{Solution:}
\]
To find the Hermitian, we take the conjugate transpose:
\[
C^H = \overline{C^T} = \begin{pmatrix} 1 – 2i & 3 + i \\ 3 – i & 2 + 4i \end{pmatrix}
\]
\[
\textbf{Q6: Find the Conjugate and Hermitian of Matrix D, where:}
\]
\[
D = \begin{pmatrix} 1 & 2 + i \\ 2 – i & 3 \end{pmatrix}
\]
\[
\textbf{Solution:}
\]
The conjugate of \( D \) is:
\[
\overline{D} = \begin{pmatrix} 1 & 2 – i \\ 2 + i & 3 \end{pmatrix}
\]
The Hermitian of \( D \) is:
\[
D^H = \overline{D^T} = \begin{pmatrix} 1 & 2 + i \\ 2 – i & 3 \end{pmatrix}
\]
\[
\textbf{Q7: Check if Matrix E is Hermitian, where:}
\]
\[
E = \begin{pmatrix} 5 + i & 2 \\ 2 & 5 – i \end{pmatrix}
\]
\[
\textbf{Solution:}
\]
\[
E^H = \overline{E^T} = \begin{pmatrix} 5 – i & 2 \\ 2 & 5 + i \end{pmatrix}
\]
Since \( E^H = E \), matrix E is Hermitian.
