Definite Bilinear Form
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A Definite Bilinear Form is a symmetric bilinear form that is also positive definite
References
2023
- (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Definite_quadratic_form#Associated_symmetric_bilinear_form Retrieved:2023-10-16.
- Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.[1] A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:
[math]\displaystyle{ \begin{align} Q(x) &= B(x, x) \\ B(x,y) &= B(y,x) = \tfrac{1}{2} [ Q(x + y) - Q(x) - Q(y) ] ~. \end{align} }[/math]
The latter formula arises from expanding [math]\displaystyle{ \; Q(x+y) = B(x+y,x+y) ~. }[/math]
- Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.[1] A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:
- ↑ This is true only over a field of characteristic other than 2, but here we consider only ordered fields, which necessarily have characteristic 0.