v2009.01.01 - Convex Optimization

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 403
−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 =
[
⎡
⎢
⎣
1
d 12 (d ]
2 12+d 13 −d 23 )
1
(d 2 12+d 13 −d 23 ) d 13
= T ∈ S 2 + (b)
1
d 12 (d 1
2 12+d 13 −d 23 ) (d ⎤
2 12+d 14 −d 24 )
1
(d 1
2 12+d 13 −d 23 ) d 13 (d 2 13+d 14 −d 34 )
1
(d 1
2 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−1
ν(i) T + (d)
d 1i
−VN TDV ⎤
N | N←N−1 ν(N)
⎦ ∈ S N−1
ν(N) T + (e)
d 1N (970)
where
⎡
ν(i) = ∆ 1 ⎢
2 ⎣
⎤
d 12 +d 1i −d 2i
d 13 +d 1i −d 3i
⎥
. ⎦ ∈ Ri−2 , i > 2 (971)
d 1,i−1 +d 1i −d i−1,i
Hence, the leading principal submatrices of EDM D must also be EDMs. 5.38
Bordered symmetric matrices in the form (970d) are known to have
intertwined [287,6.4] (or interlaced [176,4.3] [285,IV.4.1]) eigenvalues;
(confer5.11.1) that means, for the particular submatrices (970a) and (970b),
5.38 In fact, each and every principal submatrix of an EDM D is another EDM. [202,4.1]