v2010.10.26 - Convex Optimization

188 CHAPTER 2. CONVEX GEOMETRYthen we find that the pseudoinverse transpose matrixX †T =[Γ 31Γ T 3 Γ 1Γ 41Γ T 4 Γ 2](397)holds the extreme directions of the dual cone. Therefore,x = XX † x (403)is the biorthogonal expansion (395) (E.0.1), and the biorthogonalitycondition (394) can be expressed succinctly (E.1.1) 2.70X † X = I (404)Expansion w=XX † w , for any w ∈ R 2 , is unique if and only if the extremedirections of K are linearly independent; id est, iff X has no nullspace. 2.13.7.1 Pointed cones and biorthogonalityBiorthogonality condition X † X = I from Example 2.13.7.0.1 means Γ 1 andΓ 2 are linearly independent generators of K (B.1.1.1); generators becauseevery x∈ K is their conic combination. From2.10.2 we know that meansΓ 1 and Γ 2 must be extreme directions of K .A biorthogonal expansion is necessarily associated with a pointed closedconvex cone; pointed, otherwise there can be no extreme directions (2.8.1).We will address biorthogonal expansion with respect to a pointed polyhedralcone not full-dimensional in2.13.8.2.13.7.1.1 Example. Expansions implied by diagonalization.(confer6.4.3.2.1) When matrix X ∈ R M×M is diagonalizable (A.5),X = SΛS −1 = [ s 1 · · · s M ] Λ⎣⎡w T1.w T M⎤⎦ =M∑λ i s i wi T (1547)2.70 Possibly confusing is the fact that formula XX † x is simultaneously the orthogonalprojection of x on R(X) (1910), and a sum of nonorthogonal projections of x ∈ R(X) onthe range of each and every column of full-rank X skinny-or-square (E.5.0.0.2).i=1