An algebraic condition for the separation of two ellipsoids

Given two ellipsoids, we show that their characteristic equation has at least two negative roots and that the ellipsoids are separated by a plane if and only if their characteristic equation has two distinct positive roots. Furthermore, the ellipsoids touch each other externally if and only if the characteristic equation has a positive double root. An advantage of this characterization is that only the signs of the roots matter.  2001 Elsevier Science B.V. All rights reserved.