A Hermitian form on a vector space 
over the complex field 
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 dot 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 dot 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 properties (1) and (2).
