where I is the identity matrix and U* is the conjugate transpose of U. In physics, especially in quantum mechanics, the conjugate transpose of a matrix is denoted by a dagger (†) and the equation above becomes
For any unitary matrix U, the following hold:
- Given two complex vectors x and y, multiplication by U preserves their inner product; that is,
- U is normal
- U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus U has a decomposition of the form
- where V is unitary and D is diagonal and unitary.
- Its eigenspaces are orthogonal.
- For any positive integer n, the set of all n by n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
- Any square matrix with unit Euclidean norm is the average of two unitary matrices.
If U is a square, complex matrix, then the following conditions are equivalent:
- U is unitary.
- U* is unitary.
- U is invertible with U –1=U*.
- The columns of U form an orthonormal basis of with respect to the usual inner product.
- The rows of U form an orthonormal basis of with respect to the usual inner product.
- U is an isometry with respect to the usual norm.
- U is a normal matrix with eigenvalues lying on the unit circle.
- Orthogonal matrix
- Hermitian matrix
- Symplectic matrix
- Unitary group
- Special unitary group
- Unitary operator
- Matrix decomposition
- Identity matrix
- Quantum gate