v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 161is the biorthogonal expansion (336) (E.0.1), and the biorthogonalitycondition (335) can be expressed succinctly (E.1.1) 2.51X † X = I (345)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 having empty interior in2.13.8.2.13.7.1.1 Example. Expansions implied by diagonalization.(confer6.5.3.1.1) When matrix X ∈ R M×M is diagonalizable (A.5),⎡ ⎤w T1 M∑X = SΛS −1 = [s 1 · · · s M ] Λ⎣. ⎦ = λ i s i wi T (1334)coordinates for biorthogonal expansion are its eigenvalues λ i (contained indiagonal matrix Λ) when expanded in S ;⎡ ⎤wX = SS −1 1 T X M∑X = [s 1 · · · s M ] ⎣ . ⎦ = λ i s i wi T (339)wM T X i=1Coordinate value depend upon the geometric relationship of X to its linearlyindependent eigenmatrices s i w T i . (A.5.1,B.1.1)2.51 Possibly confusing is the fact that formula XX † x is simultaneously the orthogonalprojection of x on R(X) (1673), 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).w T Mi=1