v2009.01.01 - Convex Optimization

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 399
where row,column indices i,j ∈ {1... N −1}. [270] It follows:
−V T N DV N ≽ 0
D ∈ S N h
}
⎡
⇒ δ(−VNDV T N ) = ⎢
⎣
⎤
d 12
d 13
⎥
.
d 1N
⎦ ≽ 0 (959)
Multiplication of V N by any permutation matrix Ξ has null effect on its range
and nullspace. In other words, any permutation of the rows or columns of V N
produces a basis for N(1 T ); id est, R(Ξ r V N )= R(V N Ξ c )= R(V N )= N(1 T ).
Hence, −VN TDV N ≽ 0 ⇔ −VN TΞT rDΞ r V N ≽ 0 (⇔ −Ξ T c VN TDV N Ξ c ≽ 0).
Various permutation matrices 5.34 will sift the remaining d ij similarly
to (959) thereby proving their nonnegativity. Hence −VN TDV N ≽ 0 is
a sufficient test for the first property (5.2) of the Euclidean metric,
nonnegativity.
When affine dimension r equals 1, in particular, nonnegativity symmetry
and hollowness become necessary and sufficient criteria satisfying matrix
inequality (956). (6.6.0.0.1)
5.8.1.1 Strict positivity
Should we require the points in R n to be distinct, then entries of D off the
main diagonal must be strictly positive {d ij > 0, i ≠ j} and only those entries
along the main diagonal of D are 0. By similar argument, the strict matrix
inequality is a sufficient test for strict positivity of Euclidean distance-square;
−V T N DV N ≻ 0
D ∈ S N h
}
⇒ d ij > 0, i ≠ j (960)
5.34 The rule of thumb is: If Ξ r (i,1) = 1, then δ(−V T N ΞT rDΞ r V N )∈ R N−1 is some
permutation of the i th row or column of D excepting the 0 entry from the main diagonal.