📘 Chapter 5: Hermitian & Skew-Hermitian Matrices
🔍 Hermitian Matrix
A complex square matrix \( A \) is Hermitian if: \[ A^\dagger = A \] where \( A^\dagger \) is the conjugate transpose of \( A \).
This means \( A_{ij} = \overline{A_{ji}} \).
↔️ Skew-Hermitian Matrix
A complex square matrix \( A \) is Skew-Hermitian if: \[ A^\dagger = -A \] So \( A_{ij} = -\overline{A_{ji}} \) and all diagonal entries are purely imaginary or zero.
📊 Comparison
| Property | Hermitian | Skew-Hermitian |
|---|---|---|
| Transpose Rule | \( A^\dagger = A \) | \( A^\dagger = -A \) |
| Diagonal Entries | Real | Pure Imaginary or Zero |
| Eigenvalues | Real | Imaginary |
📌 Examples
Hermitian: \[ A = \begin{bmatrix} 3 & 2+i \\ 2-i & 1 \end{bmatrix} \quad \Rightarrow \quad A^\dagger = A \]
Skew-Hermitian: \[ B = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \quad \Rightarrow \quad B^\dagger = -B \]
📝 Practice
Q: Verify whether the matrix \[ C = \begin{bmatrix} 0 & 2+i \\ -2+i & 0 \end{bmatrix} \] is Hermitian, Skew-Hermitian, or neither.