Editing Unitary group (section)

{{short description|Group of unitary matrices}}
{{redirect|U(2)|other topics|U2 (disambiguation)}}
{{Group theory sidebar |Topological}}
{{Lie groups |Classical}}

In [[mathematics]], the '''unitary group''' of degree ''n'', denoted U(''n''), is the [[group (mathematics)|group]] of {{nowrap|''n''&thinsp;×&thinsp;''n''}} [[Unitary matrix|unitary matrices]], with the group operation of [[matrix multiplication]]. The unitary group is a [[subgroup]] of the [[general linear group]] {{nowrap|GL(''n'', '''C''')}}, and it has as a subgroup the [[special unitary group]], consisting of those unitary matrices with [[determinant]] 1.

In the simple case {{nowrap|1=''n'' = 1}}, the group U(1) corresponds to the [[circle group]], isomorphic to the set of all [[complex number]]s that have [[Absolute_value#Complex_numbers|absolute value]] 1, under multiplication. All the unitary groups contain copies of this group.

The unitary group U(''n'') is a [[real number|real]] [[Lie group]] of dimension ''n''<sup>2</sup>. The [[Lie algebra]] of U(''n'') consists of {{nowrap|''n''&thinsp;×&thinsp;''n''}} [[skew-Hermitian matrix|skew-Hermitian matrices]], with the [[Lie algebra|Lie bracket]] given by the [[commutator]].

The '''general unitary group''', also called the '''group of unitary similitudes''', consists of all [[matrix (mathematics)|matrices]] ''A'' such that ''A''<sup>∗</sup>''A'' 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.

Unitary groups may also be defined over fields other than the complex numbers.  The '''hyperorthogonal group''' is an archaic name for the unitary group, especially over [[finite field]]s.