📘 Chapter 6: Unitary Matrix

🔷 Definition

A complex square matrix \( U \) is unitary if: \[ U^\dagger U = U U^\dagger = I \] where \( U^\dagger \) is the conjugate transpose of \( U \).

This means that the inverse of a unitary matrix is its conjugate transpose: \( U^{-1} = U^\dagger \)

📐 Key Properties

• Preserves length: \( \|Ux\| = \|x\| \) for all vectors \( x \)
• Eigenvalues lie on the unit circle (magnitude 1)
• Generalizes orthogonal matrices to complex vector spaces

📊 Example

Let \[ U = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix} \] Then \[ U^\dagger = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -i \\ -i & 1 \end{bmatrix} \] and you can verify that: \[ U^\dagger U = I \]

📝 Practice

Q: Prove that the following matrix is unitary: \[ Q = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \] where \( \theta \in \mathbb{R} \)