Lecture 24: Advanced Linear Algebra - Unitary Linear Transformations and Unitary Matrices

Introduction

• Lecture topic: Unitary linear transformations and unitary matrices.
• Objective: To investigate linear transformations that do not change inner products and norms.

Key Concepts

Unitary Linear Transformation

• Definition: A linear transformation that preserves the norm induced by the inner product.
• Notation: A matrix is unitary if it doesn’t alter the norm induced by the inner product.
• Example: Show that a given matrix (e.g., (U = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & -1 \ 1 & 1 \end{pmatrix})) is unitary by proving (|Uv| = |v|) for any vector (v).

Properties of Unitary Matrices

1. Preservation of Norms: (|Uv| = |v|) for any vector (v).
2. Matrix Characterization:
   • If (U) is unitary, then (U^* U = I) ((U^*) is the conjugate transpose).
   • Equivalent conditions:
     • (U^* U = I)
     • (U U^* = I)
     • Preserves inner products: ((v, w) = (Uv, Uw))
     • Columns of (U) form an orthonormal basis.
     • Rows of (U) form an orthonormal basis.

Unitary vs. Invertible Matrices

• Unitary Matrices
  • Preserve the norm and inner products.
  • (U^{-1} = U^*).
  • Columns form an orthonormal basis.
• Invertible Matrices
  • Have inverse ((P^{-1})).
  • Preserve non-zeroness of vector lengths.
  • Columns form a basis (not necessarily orthonormal).

Proving Unitary Matrix Equivalences

• Proofs:
  • Show (U^*U = I) implies preservation of norms and vice versa.
  • Use properties of adjoints and matrix operations to prove equivalences.
  • Demonstrate that the orthonormality of columns implies (U^*U = I).

Examples of Unitary Matrices

Rotation Matrices

• Form: (\begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix})
• Unitary Proof: Show (U^*U = I) using trigonometry.

Reflection Matrices

• Form: (U = 2uu^* - I) (for a unit vector (u)).
• Property: Reflect across a line without changing vector lengths.

Conclusion

• Unitary Matrices: Think of them as higher-dimensional rotations and reflections.
• Characteristics: They do not stretch or deform space; they preserve inner products and norms.
• Next Lecture: Continuation into applications and further properties.