Unitary matrix

In mathematics, a complex square matrix U is unitary if its conjugate transpose U is also its inverse – that is, if where I is the identity matrix. In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (†) and the equation above becomes The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

Properties

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, . 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.

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.

Equivalent conditions

If U is a square, complex matrix, then the following conditions are equivalent:

1. U is unitary.
2. U is unitary.
3. U is invertible with U−1 = U.
4. The columns of U form an orthonormal basis of with respect to the usual inner product.
5. The rows of U form an orthonormal basis of with respect to the usual inner product.
6. U is an isometry with respect to the usual norm.
7. U is a normal matrix with eigenvalues lying on the unit circle.

Elementary constructions

2 × 2 unitary matrix

The general expression of a 2 × 2 unitary matrix is: which depends on 4 real parameters (the phase of , the phase of , the relative magnitude between and , and the angle ). The determinant of such a matrix is: The sub-group of such elements in U where 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.