We must verify linearity,symmetry and positive definiteness properties.
We can define <x,y> through matrix operations as
<x,y>=xT⋅A⋅y, x=(x1x2),xT=(x1 x2),y=(y1y2),A=(2−1−12)
and use linearity of transposition operation T and matrix multiplication
So we have
<x(1)+x(2),y>=(x(1)+x(2))T⋅A⋅y=((x(1))T+(x(2)))⋅A⋅y=((x(1))T⋅A+(x(2))T⋅A⋅y=(x(1))T⋅A⋅y+(x(2))T⋅A⋅y=<x(1),y>+<x(2),y>
and
<c⋅x,y>=(c⋅x)T⋅A⋅y=(c⋅xT)⋅A⋅y=c⋅(xT⋅A⋅y)=c⋅<x,y>
Thus linearity by the first argument of <,> is proved.
2) Symmetry property
<x,y>=xT⋅A⋅y=[xT⋅A⋅y∈R]=(xT⋅A⋅y)T=yT⋅AT⋅(xT)T=[AT=A, A−symmetric,(xT)T=x]=yT⋅A⋅x=<y,x>
From symmetry of <x,y> linearity by 2-argument is proved also.
3) Positive definiteness property.
This property better to verify by using simplest definition of <x,y>.
Let x=0 or x12+x22>0 . We have
<x,x>=2x12−2x1x2+2x22=(x1−x2)2+x12+x22>0
Thus all properties of inner product is proved.
Comments
Leave a comment