A Hermitian form on a complex vector space 
is a function 
such
that for all 
and all 
,
1. 
.
2. 
.
Here, the bar indicates the complex conjugate. It follows that
 |
(1)
|
which can be expressed by saying that 
is antilinear on the second
coordinate. Moreover, for all 
, 
, which means that 
.
An example is the Hermitian inner product of 
,
defined as
 |
(2)
|
Every Hermitian form on 
is associated with an 
Hermitian matrix 
such that
 |
(3)
|
for all row vectors 
and 
of 
. The matrix associated with the
Hermitian inner product is the 
identity matrix.
More generally, if 
is a vector space on a field 
, and 
is an automorphism such that 
, and 
, the notation 
can be used and a Hermitian form 
on 
can be defined by means of the same linearity and symmetry
properties with scalars in 
.
