Quadratic Forms

Question 1

What is Q(v) and is it linear

Answer 1

Q(v) = B(v,v) where B is a symmetric bilinear form
Not linear, Q(Lv) = B(Lv,Lv) = L^2B(v,v) = L^2Q(v)

Question 2

What is B polarisation of Q

Answer 2

B polarisation of Q is :
B(v,w) = 0.5(Q(v,w) - Q(v) - Q(w)) where Q(v) = B(v,v) symmetric bilinear

Question 3

What is the signature of Q

Answer 3

Signature of Q is the signature of its polarisation

Question 4

If Q is a quadratic form with rank r, what can be said

Answer 4

If Q is a quadratic form with rank r, it can be said
Exists basis v1,…vn of V set Q(sum xivi) = x1^2 +…+xr^2 if F = C
Q has sig(p,q) exists basis v1,…,vn s,t Q(sum xivi) = x1^2 +…+ xp^2 - xp+1^2 -…- xp+q^2