BilinearAndQuadraticForms - crowlogic/arb4j GitHub Wiki
Relationship Between Quadratic Forms and Bilinear Forms
Quadratic Forms vs Bilinear Forms in Real Vector Spaces
In real vector spaces, every quadratic form is indeed associated with a unique symmetric bilinear form.
Quadratic Form
- A quadratic form in a real vector space is a function $Q: V \to \mathbb{R}$ defined by $Q(v) = B(v, v)$ for a symmetric bilinear form $B$.
- Given a vector $x \in \mathbb{R}^n$ and a symmetric matrix $A \in \mathbb{R}^{n \times n}$, a quadratic form is an expression of the form $Q(x) = x^T A x$.
Bilinear Form
- A bilinear form $B: V \times V \to \mathbb{R}$ is a function that is linear in each of its two arguments.
- The bilinear form is symmetric if $B(u, v) = B(v, u)$ for all $u, v \in V$.
Relationship
- For a quadratic form $Q$ on a real vector space, there exists a unique symmetric bilinear form $B$ such that $Q(v) = B(v, v)$.
- This bilinear form $B$ is determined by the polarization identity:
B(u, v) = \frac{1}{2} [Q(u + v) - Q(u) - Q(v)]
Complex Vector Spaces
In complex vector spaces, the relationship between quadratic forms and bilinear forms is more nuanced due to the role of complex conjugation.
Sesquilinear Form
- In a complex vector space, a sesquilinear form is linear in its first argument and conjugate-linear in its second argument.
- A Hermitian form is a sesquilinear form satisfying $B(u, v) = \overline{B(v, u)}$.
Quadratic Forms in Complex Spaces
- Defining a quadratic form in complex spaces is less straightforward and involves a Hermitian form, but the relationship is not as direct as in real spaces.
Polarization Identity in Complex Spaces
- The polarization identity in complex spaces involves complex conjugation and works with Hermitian forms.