v2006.03.09 - Convex Optimization

250 CHAPTER 4. EUCLIDEAN DISTANCE MATRIX4.5.2 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 accomplished viaQ(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) (534)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 (534) follows directly from (460).The class of premultiplying matrices for which interpoint distances areunaffected is a little more broad than orthogonal matrices. Looking at EDMdefinition (460), it appears that any matrix Q p such thatwill have the propertyX T Q T pQ p X = X T X (535)D(Q p X) = D(X) (536)An example is skinny Q p ∈ R m×n (m>n) having orthonormal columns. Wecall such a matrix orthonormal.4.5.2.1 Inner-product form invarianceLikewise, D(Θ) (515) is rotation/reflection invariant;so (535) and (536) similarly apply.4.5.3 Invariance conclusionD(Q p Θ) = D(QΘ) = D(Θ) (537)In the making of an EDM, absolute rotation, reflection, and translationinformation is lost. Given an EDM, reconstruction of point position (4.12,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: