- 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:
“Quite generally, the familiar, just because it is familiar, is not cognitively understood. The commonest way in which we deceive either ourselves or others about understanding is by assuming something as familiar, and accepting it on that account; with all its pros and cons, such knowing never gets anywhere, and it knows not why.... The analysis of an idea, as it used to be carried out, was, in fact, nothing else than ridding it of the form in which it had become familiar.”
—Georg Wilhelm Friedrich Hegel (1770–1831)