v2007.09.17 - Convex Optimization

330 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) (794)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 (794) follows directly from (709).The class of premultiplying matrices for which interpoint distances areunaffected is a little more broad than orthogonal matrices. Looking at EDMdefinition (709), it appears that any matrix Q p such thatwill have the propertyX T Q T pQ p X = X T X (795)D(Q p X) = D(X) (796)An example is skinny Q p ∈ R m×n (m>n) having orthonormal columns. Wecall such a matrix orthonormal.5.5.2.1 Inner-product form invarianceLikewise, D(Θ) (774) is rotation/reflection invariant;so (795) and (796) similarly apply.5.5.3 Invariance conclusionD(Q p Θ) = D(QΘ) = D(Θ) (797)In the making of an EDM, absolute rotation, reflection, and translationinformation is lost. Given an EDM, reconstruction of point position (5.12,the list X) can be guaranteed correct only in affine dimension r and relativeposition. Given a noiseless complete EDM, this isometric reconstruction isunique in so far as every realization of a corresponding list X is congruent: