v2006.03.09 - Convex Optimization

324 CHAPTER 5. EDM CONED = δ ( −V DV 2) 1 1 T + 1δ ( −V DV 2) 1 T ( )− 2 −V DV12(548)By diagonalization −V DV 1 2∆= QΛQ T ∈ S N c (A.5.2) we may write( N) (∑N)∑ T∑D = δ λ i q i qiT 1 T + 1δ λ i q i qiT − 2 N λ i q i qiTi=1i=1i=1∑= N ( )λ i δ(qi qi T )1 T + 1δ(q i qi T ) T − 2q i qiTi=1(752)where q i is the i th eigenvector of −V DV 1 arranged columnar in orthogonal2matrixQ = [q 1 q 2 · · · q N ] ∈ R N×N (318)and where {δ(q i q T i )1 T + 1δ(q i q T i ) T − 2q i q T i , i=1... N} are extremedirections of some pointed polyhedral cone K ⊂ S N h and extreme directionsof EDM N . Invertibility of (752)−V DV 1 = −V ∑ N (λ2 i δ(qi qi T )1 T + 1δ(q i qi T ) T − 2q i qiTi=1∑= N λ i q i qiTi=1)V12(753)implies linear independence of those extreme directions.biorthogonal expansion is expresseddvec D = Y Y † dvec D = Y λ ( )−V DV 1 2Then the(754)whereY ∆ = [ dvec ( δ(q i q T i )1 T + 1δ(q i q T i ) T − 2q i q T i), i = 1... N]∈ R N(N−1)/2×N (755)When D belongs to the EDM cone in the subspace of symmetric hollowmatrices, unique coordinates Y † dvecD for this biorthogonal expansionmust be the nonnegative eigenvalues λ of −V DV 1 . This means D2simultaneously belongs to the EDM cone and to the pointed polyhedral conedvec K = cone(Y ).