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In abstract algebra, the unitary group of degree n over a field F (written as U(n,F)) 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).
If the field F is the field of real numbers then the unitary group coincides with the orthogonal group O(n,R), in which case the latter notation is more common. If F is the field of complex numbers one usually writes U(n) instead of U(n,C).
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
