In linear algebra, an invertible 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 conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written
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:
- Given two complex vectors x and y, multiplication by U preserves their inner product; that is, ⟨Ux, Uy⟩ = ⟨x, y⟩.
- U is normal ().
- U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, U has a decomposition of the form where V is unitary, and D is diagonal and unitary.
- Its eigenspaces are orthogonal.
- U can be written as 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).
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:
- is unitary.
- is unitary.
- is invertible with .
- The columns of form an orthonormal basis of with respect to the usual inner product. In other words, .
- The rows of form an orthonormal basis of with respect to the usual inner product. In other words, .
- is an isometry with respect to the usual norm. That is, for all , where .
- is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of ) with eigenvalues lying on the unit circle.
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 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.