v2009.01.01 - Convex Optimization

5.8. EUCLIDEAN METRIC VERSUS MATRIX CRITERIA 405
5.8.3.1.1 Example. Small completion problem, II.
Applying the inequality for λ 1 in (977) to the small completion problem on
page 348 Figure 93, the lower bound on √ d 14 (1.236 in (791)) is tightened
to 1.289 . The correct value of √ d 14 to three significant figures is 1.414 .
5.8.4 Affine dimension reduction in two dimensions
(confer5.14.4) The leading principal 2×2 submatrix T of −VN TDV N has
largest eigenvalue λ 1 (965) which is a convex function of D . 5.40 λ 1 can never
be 0 unless d 12 = d 13 = d 23 = 0. Eigenvalue λ 1 can never be negative while
the d ij are nonnegative. The remaining eigenvalue λ 2 is a concave function
of D that becomes 0 only at the upper and lower bounds of inequality (966a)
and its equivalent forms: (confer (968))
| √ d 12 − √ d 23 | ≤ √ d 13 ≤ √ d 12 + √ d 23 (a)
⇔
| √ d 12 − √ d 13 | ≤ √ d 23 ≤ √ d 12 + √ d 13 (b)
⇔
| √ d 13 − √ d 23 | ≤ √ d 12 ≤ √ d 13 + √ d 23 (c)
(978)
In between those bounds, λ 2 is strictly positive; otherwise, it would be
negative but prevented by the condition T ≽ 0.
When λ 2 becomes 0, it means triangle △ 123 has collapsed to a line
segment; a potential reduction in affine dimension r . The same logic is valid
for any particular principal 2×2 submatrix of −VN TDV N , hence applicable
to other triangles.
5.40 The largest eigenvalue of any symmetric matrix is always a convex function of its
entries, while the ⎡ smallest ⎤ eigenvalue is always concave. [53, exmp.3.10] In our particular
d 12
case, say d =
∆ ⎣d 13
⎦∈ R 3 . Then the Hessian (1638) ∇ 2 λ 1 (d)≽0 certifies convexity
d 23
whereas ∇ 2 λ 2 (d)≼0 certifies concavity. Each Hessian has rank equal to 1. The respective
gradients ∇λ 1 (d) and ∇λ 2 (d) are nowhere 0.