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)
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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)
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Every Hermitian form on is associated with an Hermitian matrix such that
(3)
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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).