Bilinear Forms

Question 1

Bilinear form

Answer 1

A bilinear form F: VxV->𝔽 is s.t.

i) F(x+y, z) = F(x,z) + F(y,z
(ii) F(λx,y)=λ(Fx,y)
(iii) F(x,y+z) = F(x,y) + F(x, z)
(iv) F(x, λy) = λF(x,y)

Question 2

symmetric

Answer 2

if F(x,y)=F(y,x)

Question 3

Definite

Answer 3

F(x,x)≠0

Question 4

Positive definite

Answer 4

F(x,x)>0

Question 5

Inner Product

Answer 5

its symmetric and positive definite

Question 6

How to write a bilinear form as a matrix product

Answer 6

F = vᵀAw

where each element of A is the coefficient of xᵢyᵢ (columns are xs and rows are ys)

Question 7

How to tell if a form is bilinear?

Answer 7

if you can put it into the matrix form F = vᵀAw it is bilinear

Question 8

quadratic form

Answer 8

a polynomial in several variables where each term has total degree 2
e.g. x²+y²+xy

Question 9

Sequelinear form

Answer 9

A sesquelinear form F: VxV->𝔽 is s.t.

i) F(x+y, z) = F(x,z) + F(y,z
(ii) F(λx,y)=λ*(Fx,y)
(iii) F(x,y+z) = F(x,y) + F(x, z)
(iv) F(x, λy) = λF(x,y)

Question 10

Conjugate symmetric

Answer 10

if F(x,y) = F(y,x)*

Question 11

conjugate symmetric sequelinear form inner product

Answer 11

conjugate symmetric sequelinear form is an inner product if it is positive definite

Question 12

F(v,w) =

Answer 12

[|v|]ᵀ[|F|][|w|]

in sequeliniear form put conjugate wherever there is a transpose

Question 13

if you change basis for a bilinear form matrix representation

Answer 13

[|F|]’ = Pᵀ[|F|]P

in sequeliniear form put conjugate wherever there is a transpose

Question 14

orthogonal vectors

Answer 14

vectors are perpendicular

Question 15

How to find a basis of V s.t. [|F|] is diagonal

Answer 15

NOTE F MUST BE DEFINITE

1. take the first element, f₁, of the original basis and let it be e₁ (the first element of the new basis)
2. calculate F(e₁,e₁) , F(e₁,fᵢ) for all i
3. fᵢ= fᵢ - F(e₁,e₁)/F(e₁,fᵢ) *e₁
4. choose e₂ = next fᵢ
5. check that e₁,e₂ are orthogonal i.e. F(e₁,e₂)=0
6. calculate F(e₂, e₂). Repeat the process.

Question 16

To see if F is positive definite or not:

Answer 16

• find a counter example

- if v = v₁e₁ + v₂e₂, calculate F(v,v). If =0 only if v₁=v₂=0 then positive definite.

Question 17

How to normalise

Answer 17

divide by sqrt (F(eᵢ,eᵢ))

Question 18

How to find a basis of V s.t. [|F|] is diagonal and F(eⱼ,eⱼ))=0 (F is not definite)

Answer 18

• chose eⱼ to be a different original vector where F(eⱼ,eⱼ)≠0
• If F(vᵢ,vᵢ)=0 but F(vᵢ,vⱼ)≠0 then take eⱼ = vᵢ + vⱼ. Then delete one of the original vectors and return the other to use again.
• If F(vᵢ,vⱼ)=0 for all i,j then eⱼ=vⱼ for all remaining vⱼs

Question 19

rank

Answer 19

The sum k+m where k is the number of positive terms on the diagonal of [|F|] and m is the number of negative terms on the diagonal of [|F|]

Question 20

signature

Answer 20

The difference k-m where k is the number of positive terms on the diagonal of [|F|] and m is the number of negative terms on the diagonal of [|F|]