The unitary group 
is the group of 
unitary matrices,
 |
(1)
|
where 
is the conjugate transpose and 
is the identity matrix.
Equivalently, it is the group of complex linear
transformations that are isometries of the standard
Hermitian inner product on 
.
The group 
is a connected compact
Lie group of real dimension 
. Its Lie algebra is the real
vector space of antihermitian
matrices,
 |
(2)
|
where 
denotes the set of all 
complex matrices. The determinant
maps 
onto 
,
and its kernel is the special
unitary group. This gives the exact sequence
 |
(3)
|
The group center is the subgroup of scalar matrices 
with 
.
More generally, a unitary group is the group of linear transformations preserving a nonsingular Hermitian
form. Indefinite Hermitian forms over 
give the matrix
groups 
.
Over the finite field 
, the full group of isometries
is denoted 
in the Atlas of Finite Groups and is described in the general
unitary group entry. The notation 
is convention-dependent and is also used in the Atlas
for the simple group 
.
