A positive semi-definite matrix has all non-negative eigenvalues. The proof is as follows. Letvbe an eigenvector ofAwith eigenvalueλsoAv=λv.BecauseAis positive semi-definite we havehv, Avi ≥0.Using the fact thatvis an eigenvector we also havehv, Avi=hv, λvi=λhv,vi.Combining these two results and remembering thathv,vi>0 for an eigenvector, we have ourresult:λ≥0.IV.8.3. Reconstructing points inRpWe begin with a simple example, taking our objects to be a set ofnpointsw1,w2, . . . ,wninRp.We take the dissimilarity between objectsiandjto be the usual distance between those pointsTij=kwi-wjk.Is it possible to reconstruct the relative location of the points from the data inTalone?It is important to notice that any reconstruction that we find is not unique. We are free to rotate,reflect and translate the points and they still satisfy the only requirement that we make of them,namely that the distances between the points are as specified.Reconstructing two points inRpConsider two points,w1andw2, that are a known distance, say 2, apart. A single line can alwaysbe placed between any two points and therefore we expect that the points can be represented inone dimension. If we only know the distance between the points, then a possible representation ofthem isv1=0andv2=2 .Reconstructing three points inRpNow consider three points,w1,w2andw3with dissimilarity matrix:T=035304540.206