Classification Of Symmetric Bilinear Forms

Question 1

What is symmetric bilinear form B if B(v,v) = 0

Answer 1

If B(v,v) = 0, then B identical to 0
B(v,v) + B(v,w) + B(w,v) + B(w,w) = 2B(v,w) and 2 =/ 0

Question 2

What does the diagonalisable theorem state

Answer 2

Diagonalisation theorem states that:
Can find basis v1,…,vn of V s.t
B(vi,vj) = 0 if I=/ j I.e B has diagonal matrix w.r.t basis
Call v1,…,vn diagonalising basis

Question 3

If A is a square symmetric matrix, then what diagonal matrix can be mad

Answer 3

If A is a square symmetric matrix then P^TAP is diagonal

Question 4

Find diagonalising basis Q9 exams

Answer 4

LEARN HOW TO FIND DIAGONALISING BASIS Q9 EXAMS

Question 5

What is Sylvester’s theorem

Answer 5

Sylvester’s theorem is:
If B is a symmetric bilinear form of sig(p,q), then:
P + q = rank B
Any diagonal matrix representing B has p positive entries and q negative entries

Question 6

When is a symmetric bilinear form B positive/negative definite

Answer 6

Symmetric bilinear form B is positive definite on V if B(v,v) > 0 for all v in V\0 and negative definite if -B is positive definite

Question 7

What is signature of B

Answer 7

Signature of B is (p,q) where:
P = max(dim U | U <= V and B is positive definite) (No of positive entries)
Q = max(dimW| W <= V and B is neg definite (no of negative entries)

Question 8

When is B positive definite

Answer 8

B is positive definite when B is a real inner product