Synmetric Bilinear Forms

Question 1

When are matrices A,B congruent

Answer 1

Matrices A,B are congruent if exists invertible matrix P s.t B = P^TAP

Question 2

When is a Bilinear form B symmetric and example

Answer 2

Bilinear form B is symmetric when:
B(v,w) = B(w,v)
Symmetric iff it’s matrix is symmetric w.r.t to a basis
E.g real inner products

Question 3

What is rad B of B and when is B non-degenerate B: V x V to F

Answer 3

Rad B of B is:
Rad B = (v in V | B(v,w) = 0 for all w in V) <= V
B is non-degenerate when rad B = 0

Question 4

What is rank B

Answer 4

Rank B = dim V - dim B
Non-degenerate is rank B = dim V

Question 5

What is an inner product

Answer 5

Inner product is non-deg as inner(v,v) > 0

Question 6

How understand rank and radicals

Answer 6

Understand rank and radicals by:
B defines map B : V to V* by
B(v)(w) = B(v,w)
Then B(v) in V* since B is linear in 2nd slot
B: V to V* is linear map since B linear in 1st slot
Ker B = ( v in V | B(v) = 0) = (v in V | B(v)(w) = 0 for all w in V) = rad B

Question 7

When is B non-deg (matrices)

Answer 7

B is non-deg when B injects or when V is finite dimensional, an isomorphism
And is non-deg iff det A =/ 0 where A is matrix representing B

Question 8

If Bilinear map B has matrix A, then what is A

Answer 8

If bilinear map B has matrix A,
A is B(Vj)(Vi) = B(Vj,Vi) = Aji = Aij where vi a are basis vectors

Question 9

What is rad B(A) (matrix A representing bilinear map B)

Answer 9

Rad B(A) = ker A

Question 10

When is B non-deg and what if B is symmetric bilinear form

Answer 10

B is non-deg iff det A =/ 0, if B is symmetric bilinear form, rank B = rank A