- The term Hermitian form may also refer to a different concept than that explained below: it may refer to a certain differential form on a Hermitian manifold.
A Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V × V → C such that
The standard Hermitian form on Cn is given by
More generally, the inner product on any complex Hilbert space is a Hermitian form.
A vector space with a Hermitian form (V,h) is called a Hermitian space.
If V is a finite-dimensional space, then relative to any basis {ei} of V, a Hermitian form is represented by a Hermitian matrix H:
The components of H are given by Hij = h(ei, ej).
The quadratic form associated to a Hermitian form
- Q(z) = h(z,z)
is always real. Actually one can show that a sesquilinear form is Hermitian iff the associated quadratic form is real for all z ∈ V.
Read more about this topic: Sesquilinear Form
Famous quotes containing the word form:
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