The function defined by I(x,y)=xTAy is a symmetric bilinear function
1) it satisfies the axiom <u,v> = <v,u> in such:
⟨x,y⟩=xT Ay=x⋅(Ay)
Since A is symmetric
⟨x,y⟩=x⋅(Ay)=(Ay)⋅x
=(Ay)Tx=yT AT x=yT Ax=⟨y,x⟩.
(ii) For any vector x,y,z and any real number r, we have:
⟨rx,y⟩=(rx)TAy=rxTAy=r⟨x,y⟩
and
⟨x+y,z⟩=(x+y)TAz=(xT+yT)Az
=xTAz+yTAz=⟨x,y⟩=⟨y,z⟩
Therefore the linearity in the first argument is satisfied
It satisfies the rule of positive- Definiteness in such
If x=0 then we have
⟨x,x⟩=xTAx>0
Examples of matrices A for which the corresponding I is not an inner product
→ A matrix of order m×n
→ A matrix A =AT
→ A matrix A whose eigenvalues λ≤0
→ A matrix A whose determinant det A=0
Properties of A which would make the corresponding I on inner product
Matrix A should be a real symmetric n×n matrix
Matrix A must be positive definite in that
xTAx>0
Matrix A has real eigenvalues
The eigenvectors of matrix A corresponding to the eigenvalues are orthogonal
Matrix A is diagonalizable
Part 6(B) sections
I is inner product if it satisfies conjugate symmetry, linearity and positive Definiteness axioms
(I) <q(x),p(x)>=q(0)p(0)+q(21)q(21)+q(1)p(1)
=p(0)q(0)+p(21)q(21)+p(1)q(1)
∴<q(x),p(x)>=<p(x),q(x)>
I is symmetric
(ii) c<p,q>=c[p(0)q(0)+p(21q(21+p(1)q(1)]
=cp(0)q(0)+cp(21)q(21)+cp(1)q(1)
=<cpf,q>
I satisfies linearity
(iii) <p,p>=p(0)p(0)+p(21)p(21)+p(1)p(1)>0
If p(x) is not zero
Part 6(B) section 2
The standard polynomial basis for R[x]deg≤2 is defined [1,x,x2]
Let orthogonal basis of I be (b1,b2,b3)
b1=1
b2=x−proj1x
∴b2=x−<1,1><1,x>1
<1,x>=(1)(0)+(1)(21)+(1)(1)=23
<1,1>=(1)(1)+(1)(1)+(1)(1)=3
b2=x−21
b3=x2−<1,1><x2,1>1− <x−21,x−21><x2,x−21>(x−21)
<x2,1>=(1)(0)+(1)(21)2+(1)(1)2=45
<x−21,x−21>=(2−1)(2−1)+(0)(0)+(21)(21)=21
<x2,x−21>=(0)(2−1)+(21)2(0)+(1)2(21)
=21
b3=x2−125−1(x−21)
=x2−x+121
Comments