What are the properties of unitary transform

Unitary transforms are mathematical operations that preserve the inner product and the norm of vectors in a vector space. Here are the key properties of unitary transforms:

1. **Preservation of Inner Product**: A unitary transform \( U \) between two vector spaces \( V \) and \( W \) preserves the inner product. For any vectors \( \mathbf{u}, \mathbf{v} \in V \), it holds that \( \langle U\mathbf{u}, U\mathbf{v} \rangle = \langle \mathbf{u}, \mathbf{v} \rangle \).

2. **Orthogonality Preservation**: Unitary transforms preserve orthogonality between vectors. If \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal in \( V \), then \( U\mathbf{u} \) and \( U\mathbf{v} \) are orthogonal in \( W \).

3. **Norm Preservation**: The norm (or length) of vectors is preserved under unitary transforms. Specifically, \( \| U\mathbf{u} \| = \| \mathbf{u} \| \) for any vector \( \mathbf{u} \in V \).

4. **Invertibility**: Unitary transforms are invertible. There exists a unique inverse transform \( U^{-1} \) such that \( U^{-1}U = UU^{-1} = I \), where \( I \) is the identity operator.

5. **Eigenvalue Properties**: Unitary transforms have eigenvalues with magnitudes equal to 1. Specifically, if \( U\mathbf{u} = \lambda \mathbf{u} \) for some eigenvalue \( \lambda \), then \( |\lambda| = 1 \).

6. **Matrix Representation**: In terms of matrices, a unitary matrix \( U \) satisfies \( U^\dagger U = UU^\dagger = I \), where \( U^\dagger \) denotes the conjugate transpose (or Hermitian transpose) of \( U \).

Unitary transforms are fundamental in various areas of mathematics and physics, including quantum mechanics, signal processing, and linear algebra, due to their ability to preserve geometric properties such as angles and lengths in vector spaces.