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Unitary matrices

Created over 8 years ago, updated 10 days ago

Definition: A complex square matrix $U$ is unitary if its conjugate transpose equals its inverse:

$$U^* U = UU^* = I \quad \text{or equivalently} \quad U^{-1} = U^*$$

where $U^* = (\overline{U})^T$ is the conjugate transpose.

Example:
$$U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$$

Properties:

  • $|\det(U)| = 1$
  • Preserves inner products: $\langle Ux, Uy \rangle = \langle x, y \rangle$
  • Preserves norms: $\|Ux\| = \|x\|$
  • Eigenvalues lie on the unit circle ($|\lambda| = 1$)
  • Columns (and rows) form an orthonormal basis
  • The product of unitary matrices is unitary

Unitary matrices are the complex analog of orthogonal matrices. They represent isometries (distance-preserving transformations) in complex vector spaces and are fundamental in quantum mechanics.