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 
.
