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.
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.
  4. The columns of U form an orthonormal basis of with respect to the usual inner product. In other words, U∗U =I.
  5. The rows of U form an orthonormal basis of with respect to the usual inner product. In other words, U U∗ = I.
  6. U is an isometry with respect to the usual norm. That is, for all, where.
  7. U is a normal matrix with eigenvalues lying on the unit circle.

    Elementary constructions

2 × 2 unitary matrix

The general expression of a unitary matrix is
which depends on 4 real parameters. The determinant of such a matrix is
The sub-group of those elements with is called the special unitary group SU.
The matrix can also be written in this alternative form:
which, by introducing and, takes the following factorization:
This expression highlights the relation between unitary matrices and orthogonal matrices of angle.
Another factorization is
Many other factorizations of a unitary matrix in basic matrices are possible.