v2010.10.26 - Convex Optimization

436 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.5.2 Rotation/ReflectionRotation of the list X ∈ R n×N about some arbitrary point α∈ R n , orreflection through some affine subset containing α , can be accomplishedvia Q(X −α1 T ) where Q is an orthogonal matrix (B.5).We rightfully expectD ( Q(X − α1 T ) ) = D(QX − β1 T ) = D(QX) = D(X) (978)Because list-form D(X) is translation invariant, we may safely ignoreoffset and consider only the impact of matrices that premultiply X .Interpoint distances are unaffected by rotation or reflection; we say,EDM D is rotation/reflection invariant. Proof follows from the fact,Q T =Q −1 ⇒ X T Q T QX =X T X . So (978) follows directly from (891).The class of premultiplying matrices for which interpoint distances areunaffected is a little more broad than orthogonal matrices. Looking at EDMdefinition (891), it appears that any matrix Q p such thatX T Q T pQ p X = X T X (979)will have the propertyD(Q p X) = D(X) (980)An example is skinny Q p ∈ R m×n (m>n) having orthonormal columns. Wecall such a matrix orthonormal.5.5.2.0.1 Example. Reflection prevention and quadrature rotation.Consider the EDM graph in Figure 128b representing known distancebetween vertices (Figure 128a) of a tilted-square diamond in R 2 . Supposesome geometrical optimization problem were posed where isometrictransformation is allowed excepting reflection, and where rotation mustbe quantized so that only quadrature rotations are allowed; only multiplesof π/2.In two dimensions, a counterclockwise rotation of any vector about theorigin by angle θ is prescribed by the orthogonal matrix[ ]cos θ − sin θQ =sin θ cosθ(981)