v2010.10.26 - Convex Optimization

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 451where row,column indices i,j ∈ {1... N −1}. [312] It follows:−V T N DV N ≽ 0D ∈ S N h}⎡⇒ δ(−VNDV T N ) = ⎢⎣⎤d 12d 13⎥.d 1N⎦ ≽ 0 (1054)Multiplication of V N by any permutation matrix Ξ has null effect on its rangeand nullspace. In other words, any permutation of the rows or columns of V Nproduces 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 5.37 will sift the remaining d ij similarlyto (1054) thereby proving their nonnegativity. Hence −VN TDV N ≽ 0 isa sufficient test for the first property (5.2) of the Euclidean metric,nonnegativity.When affine dimension r equals 1, in particular, nonnegativity symmetryand hollowness become necessary and sufficient criteria satisfying matrixinequality (1051). (6.5.0.0.1)5.8.1.1 Strict positivityShould we require the points in R n to be distinct, then entries of D off themain diagonal must be strictly positive {d ij > 0, i ≠ j} and only those entriesalong the main diagonal of D are 0. By similar argument, the strict matrixinequality is a sufficient test for strict positivity of Euclidean distance-square;−V T N DV N ≻ 0D ∈ S N h5.8.2 Triangle inequality property 4}⇒ d ij > 0, i ≠ j (1055)In light of Kreyszig’s observation [227,1.1 prob.15] that properties 2through 4 of the Euclidean metric (5.2) together imply nonnegativityproperty 1,5.37 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 somepermutation of the i th row or column of D excepting the 0 entry from the main diagonal.