In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if
where I is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (
), so the equation above is written
A complex matrix U is special unitary if it is unitary and its matrix determinant equals 1.
For real numbers, the 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.
For any unitary matrix U of finite size, the following hold: For any nonnegative integer n, the set of all n × n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
Every square matrix with unit Euclidean norm is the average of two unitary matrices.
[1] If U is a square, complex matrix, then the following conditions are equivalent:[2] One 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 form is configured so the determinant of such a matrix is
The sub-group of those elements
is called the special unitary group SU(2).
Among several alternative forms, the matrix U can be written in this form:
cos θ
sin θ
φ , α , β , θ
α = ψ + δ
β = ψ − δ
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.
cos ρ
− sin ρ
sin ρ
cos ρ
cos σ
sin σ
− sin σ
cos σ
Many other factorizations of a unitary matrix in basic matrices are possible.