How Do You Maximize a Quadratic Form on a Sphere with an Affine Transformation?

[Highwind]

Hi,

I search for the maximum of a quadric for points on a sphere.
I have an affine transform A (4x4 matrix, in homogeneous coord.) and apply it to points on (and inside) a sphere x \in S_{m,r} \Leftrightarrow (x-m)^2<=r^2. (Although I think the extremum must be on the surface of the sphere?).

Now I want to find the maximum displacement of any point in/on S:
d^2 (x) = (A x - x)^2 = ( (A-E) x)^2 = x^T (A-E)^T (A-E) x

The matrix B:= (A-E)^T (A-E) is of course symmetric.

so what is:
max_{x \in S} \ d^2(x) = max_{x \in S} \ x^T B x

Thanks for any help...

 

[fresh_42]

This cannot be answered in this generality. E.g. $B$ could be negative (or positive or neither) definit, and you have basically only a symmetric bilinear form. If it is, then $0$ will be the maximum.