BilinearAndQuadraticForms

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.