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 ([dagger](https://bliptext.com/articles/dagger-mark)), 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.

Properties

For any unitary matrix U of finite size, the following hold: x and y , multiplication by U preserves their inner product; that is, ⟨Ux, Uy⟩ = ⟨x, y⟩ . U = eiH , where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix. 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.

Equivalent conditions

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

Elementary constructions

2 × 2 unitary matrix

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 \ U\ with is called the special unitary group SU(2). Among several alternative forms, the matrix U can be written in this form: where and above, and the angles can take any values. By introducing and has the following factorization: This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ. Another factorization is Many other factorizations of a unitary matrix in basic matrices are possible.