where I is the identity matrix and U* is the conjugate transpose of U. In physics, especially in quantum mechanics, the Hermitian conjugate 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 nonnegative 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.
- U can be written as U=eiH where e indicates matrix exponential, i is the imaginary unit and H is an Hermitian matrix.
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.
2x2 Unitary matrix
The general expression of a 2x2 unitary matrix is:
which depends on 4 real parameters. The determinant of such a matrix is:
If φ=0, the group created by U is called special unitary group SU(2).
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 2x2 unitary matrices and 2x2 orthogonal matrices of angle θ.
Many other factorizations of a unitary matrix in basic matrices are possible.
3x3 Unitary matrix
The general expression of 3x3 unitary matrix is:
where φn, n=1,...,5 are arbitrary real numbers, while K is the Cabibbo–Kobayashi–Maskawa matrix.
- Orthogonal matrix
- Hermitian matrix
- Symplectic matrix
- Unitary group
- Special unitary group
- Unitary operator
- Matrix decomposition
- Identity matrix
- Quantum gate