Ellipsoid algebra: converting forms

[d_forage]

I have a matrix D (it happens to be in R^(nxm) where n>>m, but I don't think that is relevant at this point). I also have a vector t in R^n.

I am interested in rewriting the set

{x | (Dx-t)'(Dx-t) <= c} in standard ellipsoid form: --> {x | (x-z)'E(x-z)<=b} where E is an mxm positive semi definite matrix. So, I'd like to write z, E and b in terms of D, t, and c.

Is there an algebraic way to do this?

(The application area is radiation therapy, D represents the so-called dose-influence matrix.)

 

[I like Serena]

Welcome to PF, d_forage!

Here's one thing you can do.

You can rewrite your expression Dx-t in homogeneous coordinates (see http://en.wikipedia.org/wiki/Transformation_matrix#Affine_transformations").
This means that it becomes D.x-t = F.<x,1>
And (Dx-t)'(Dx-t) = <x,1>'F'F<x,1> = <x,1>'G<x,1>

Since G=F'F, G will be a symmetric matrix which can be diagonalized.