For any unitary matrix U of finite size, the following hold:
- Given two complex vectors x and y, multiplication by U preserves their inner product; that is, ⟨Ux, Uy⟩ = ⟨x, y⟩.
- 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.
- U can be written as U = eiH, where e indicates matrix exponential, i is the imaginary unit, and H is a Hermitian matrix.
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.
2 × 2 unitary matrix
The general expression of a 2 × 2 unitary matrix is:
which depends on 4 real parameters (the phase of a, the phase of b, the relative magnitude between a and b, and the angle φ). The determinant of such a matrix is:
The sub-group of such elements in U where det(U) = 1 is called the special unitary group SU(2).
The matrix U can also be written in this alternative form:
which, by introducing φ1 = ψ + Δ and φ2 = ψ − Δ, takes the following factorization:
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.
Many other factorizations of a unitary matrix in basic matrices are possible.
- Orthogonal matrix
- Hermitian matrix
- Symplectic matrix
- Orthogonal group O(n)
- Special Orthogonal group SO(n)
- Unitary group U(n)
- Special Unitary group SU(n)
- Unitary operator
- Matrix decomposition
- Quantum gate