v2006.03.09 - Convex Optimization

4.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 267−V T N DV N | N←1 = [ ∅ ] (o)−V T N DV N | N←2 = [d 12 ] ∈ S + (a)−V T N DV N | N←3 =−V T N DV N | N←4 =[⎡⎢⎣1d 12 (d ]2 12+d 13 −d 23 )1(d 2 12+d 13 −d 23 ) d 13= T ∈ S 2 + (b)1d 12 (d 12 12+d 13 −d 23 ) (d ⎤2 12+d 14 −d 24 )1(d 12 12+d 13 −d 23 ) d 13 (d 2 13+d 14 −d 34 )1(d 12 12+d 14 −d 24 ) (d 2 13+d 14 −d 34 ) d 14⎥⎦ (c).−VN TDV N | N←i =.−VN TDV N =⎡⎣⎡⎣−VN TDV ⎤N | N←i−1 ν(i)⎦ ∈ S i−1ν T + (d)(i) d 1i−VN TDV ⎤N | N←N−1 ν(N)⎦ ∈ S N−1ν T + (e)(N) d 1N (613)where⎡ν(i) = ∆ 1 ⎢2 ⎣⎤d 12 +d 1i −d 2id 13 +d 1i −d 3i⎥. ⎦ ∈ Ri−2 , i > 2 (614)d 1,i−1 +d 1i −d i−1,iHence, the leading principal submatrices of EDM D must also be EDMs. 4.31Bordered symmetric matrices in the form (613d) are known to haveintertwined [212,6.4] (or interlaced [125,4.3] [209,IV.4.1]) eigenvalues;(confer4.11.1) that means, for the particular submatrices (613a) and (613b),4.31 In fact, each and every principal submatrix of an EDM D is another EDM. [145,4.1]