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In mathematics, the unitary group of degree n, denoted U, is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL. Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.
In the simple case n = 1, the group U corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication. All the unitary groups contain copies of this group.
The unitary group U is a real Lie group of dimension n. The Lie algebra of U consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator.
The general unitary group consists of all matrices A such that AA is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.