Unitary matrix

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.