Quadratic form

A quadratic form over a field $𝕂$ is a map $𝑞 : 𝑉 \to 𝕂$ from a finite vector space $𝕂$ such that

𝑞 (𝜆 𝑥) = 𝜆 2 𝑞 (𝑥)

for all $𝑥 \in 𝑉$ and the corresponding polar form $𝑏 𝑞 (𝑢, 𝑣) = 𝑞 (𝑢 + 𝑣) - 𝑞 (𝑢) - 𝑞 (𝑣)$ is a bilinear form. #m/def/linalg A vector space equipped with a quadratic form is a Quadratic space. Equivalently, $𝑞$ is an algebraic form of degree 2

𝑞 (𝑥 1, \dots, 𝑥 𝑛) = 𝑛 \sum 𝑖 = 1 𝑛 \sum 𝑗 = 1 𝑎 𝑖 𝑗 𝑥 𝑖 𝑥 𝑗

or using a matrix $𝐴 = (𝑎 𝑖 𝑗)$

𝑞 (𝑥) = 𝑥 𝖳 𝐴 𝑥

A space equipped with a quadratic form is called a quadratic space.

Further terminology

See Further terminology

Properties

Correspondence between quadratic forms and symmetric bilinear forms away from 2
Correspondence between quadratic forms and alternating bilinear forms at 2
Diagonalization of a quadratic form away from 2

Projective quadric

#state/tidy | #lang/en | #SemBr

Quadratic form

Further terminology

Properties

Related