v2007.09.17 - Convex Optimization

122 CHAPTER 2. CONVEX GEOMETRY2.10.0.0.2 Exercise. Conically independent columns and rows.We suspect the number of conically independent columns (rows) of X tobe the same for X †T , where † denotes matrix pseudoinverse (E). Provewhether it holds that the columns (rows) of X are c.i. ⇔ the columns (rows)of X †T are c.i.2.10.1 Preservation of conic independenceIndependence in the linear (2.1.2.1), affine (2.4.2.4), and conic senses canbe preserved under linear transformation. Suppose a matrix X ∈ R n×N (240)holds a conically independent set columnar. Consider the transformationT(X) : R n×N → R n×N ∆ = XY (242)where the given matrix Y = ∆ [y 1 y 2 · · · y N ]∈ R N×N is represented by linearoperator T . Conic independence of {Xy i ∈ R n , i=1... N} demands, bydefinition (239),Xy i ζ i + · · · + Xy j ζ j − Xy l ζ l = 0, i≠ · · · ≠j ≠l = 1... N (243)have no nontrivial solution ζ ∈ R N + . That is ensured by conic independenceof {y i ∈ R N } and by R(Y )∩ N(X) = 0 ; seen by factoring X .2.10.1.1 linear maps of cones[18,7] If K is a convex cone in Euclidean space R and T is any linearmapping from R to Euclidean space M , then T(K) is a convex cone in Mand x ≼ y with respect to K implies T(x) ≼ T(y) with respect to T(K).If K is closed or has nonempty interior in R , then so is T(K) in M .If T is a linear bijection, then x ≼ y ⇔ T(x) ≼ T(y). Further, if F isa face of K , then T(F) is a face of T(K).2.10.2 Pointed closed convex K & conic independenceThe following bullets can be derived from definitions (155) and (239) inconjunction with the extremes theorem (2.8.1.1.1):The set of all extreme directions from a pointed closed convex coneK ⊂ R n is not necessarily a linearly independent set, yet it must be a conicallyindependent set; (compare Figure 15 on page 60 with Figure 38(a))