In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group Gl(n,F).

In the simple case n = 1, the group U(1) is the unit circle in the complex plane, under multiplication. All the complex unitary groups contain copies of this group.

If the field F is the field of real numbers then the unitary group coincides with the orthogonal group O(n,R). If F is the field of complex numbers one usually writes U(n) for the unitary group of degree n.

The unitary group U(n) is a real Lie group of dimension n². The Lie algebra of U(n) consists of complex n-by-n Skew-hermitian matrices, with the Lie bracket given by the commutator.

See also: Special unitary group