
56
B.L.R. De Moor
3.1 GEOMETRIC INTERPRETATION
Consider the two quadratic surfaces
xTcx
=
1 and
xTx
= 1. While the second surface is
the unit sphere in p dimensions, the first surface can come in many disguises. For instance,
when C is positive definite, it is an ellipsoid. In three dimensions, depending on its inertia,
it can be a one-sheeted or two-sheeted hyperboloid or a (hyperbolic) cilinder. In higher
dimensions, there are many possibilities, the enumeration of which is not relevant right
now. In each of these cases, the vectors Cx and x are the normal vectors at x to the two ...