In linear algebra, a complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if
where I 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.
For any unitary matrix U of finite size, the following hold:
- where V is unitary, and D is diagonal and unitary.
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.
If U is a square, complex matrix, then the following conditions are equivalent:
- U is unitary.
- U* is unitary.
- U is invertible with .
- The columns of U form an orthonormal basis of with respect to the usual inner product. In other words, U*U =I.
- The rows of U form an orthonormal basis of with respect to the usual inner product. In other words, U U* = I.
- U is an isometry with respect to the usual norm. That is, for all , where .
- U is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of U) with eigenvalues lying on the unit circle.
2 × 2 unitary matrix
The general expression of a 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 ?). The determinant of such a matrix is
The sub-group of those elements with is called the special unitary group SU(2).
The matrix U 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.