v2010.10.26 - Convex Optimization

488 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.14.2.1 Nonnegative determinantBy (962) when D ∈EDM 4 , −VN TDV N is equal to inner product (957),⎡√ √ ⎤d 12 d12 d 13 cos θ 213 d12 √d12 √d 14 cos θ 214Θ T Θ = ⎣ d 13 cosθ 213 d 13 d13 √d12d 14 cos θ 314⎦√d 14 cosθ 214 d13 d 14 cos θ 314 d 14(1157)Because Euclidean space is an inner-product space, the more conciseinner-product form of the determinant is admitted;det(Θ T Θ) = −d 12 d 13 d 14(cos(θ213 ) 2 +cos(θ 214 ) 2 +cos(θ 314 ) 2 − 2 cos θ 213 cosθ 214 cosθ 314 − 1 )The determinant is nonnegative if and only if(1158)cos θ 214 cos θ 314 − √ sin(θ 214 ) 2 sin(θ 314 ) 2 ≤ cos θ 213 ≤ cos θ 214 cos θ 314 + √ sin(θ 214 ) 2 sin(θ 314 ) 2⇔cos θ 213 cos θ 314 − √ sin(θ 213 ) 2 sin(θ 314 ) 2 ≤ cos θ 214 ≤ cos θ 213 cos θ 314 + √ sin(θ 213 ) 2 sin(θ 314 ) 2⇔cos θ 213 cos θ 214 − √ sin(θ 213 ) 2 sin(θ 214 ) 2 ≤ cos θ 314 ≤ cos θ 213 cos θ 214 + √ sin(θ 213 ) 2 sin(θ 214 ) 2which simplifies, for 0 ≤ θ i1l ,θ l1j ,θ i1j ≤ π and all i≠j ≠l ∈{2, 3, 4} , to(1159)cos(θ i1l + θ l1j ) ≤ cos θ i1j ≤ cos(θ i1l − θ l1j ) (1160)Analogously to triangle inequality (1073), the determinant is 0 upon equalityon either side of (1160) which is tight. Inequality (1160) can be equivalentlywritten linearly as a triangle inequality between relative angles [393,1.4];|θ i1l − θ l1j | ≤ θ i1j ≤ θ i1l + θ l1jθ i1l + θ l1j + θ i1j ≤ 2π0 ≤ θ i1l ,θ l1j ,θ i1j ≤ π(1161)