Homogenous polynomial

Quadratic form

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

𝑞(𝜆𝑥)=𝜆2𝑞(𝑥)

for all 𝑥 ∈𝑉 and the corresponding polar form 𝑏𝑞(𝑢,𝑣) =𝑞(𝑢 +𝑣) −𝑞(𝑢) −𝑞(𝑣) is a bilinear form. linalg A vector space equipped with a quadratic form is a Quadratic space. Equivalently, 𝑞 is an algebraic form of degree 2

𝑞(𝑥1,…,𝑥𝑛)=𝑛∑𝑖=1𝑛∑𝑗=1𝑎𝑖𝑗𝑥𝑖𝑥𝑗

or using a matrix 𝐴 =(𝑎𝑖𝑗)

𝑞(𝑥)=𝑥𝖳𝐴𝑥

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

Related

• Projective quadric

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