Unitary matrix

In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
where is the identity matrix.
In physics, especially in quantum mechanics, the Hermitian adjoint 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 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").
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:
Elementary constructions.
2 × 2 unitary matrix.
A factorization is
Many other factorizations of a unitary matrix in basic matrices are possible.