- 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:
“Women over fifty already form one of the largest groups in the population structure of the western world. As long as they like themselves, they will not be an oppressed minority. In order to like themselves they must reject trivialization by others of who and what they are. A grown woman should not have to masquerade as a girl in order to remain in the land of the living.”
—Germaine Greer (b. 1939)