v2010.10.26 - Convex Optimization

190 CHAPTER 2. CONVEX GEOMETRYmust be its nonnegative eigenvalues (1450) when expanded in Q ; id est, forX ∈ S M +M∑M∑X = QQ T X = q i qi T X = λ i q i qi T ∈ S M + (401)i=1where λ i ≥0 is the i th eigenvalue of X . This means matrix Xsimultaneously belongs to the positive semidefinite cone and to the pointedpolyhedral cone K formed by the conic hull of its eigenmatrices. 2.13.7.1.2 Example. Expansion respecting nonpositive orthant.Suppose x ∈ K any orthant in R n . 2.71 Then coordinates for biorthogonalexpansion of x must be nonnegative; in fact, absolute value of the Cartesiancoordinates.Suppose, in particular, x belongs to the nonpositive orthant K = R n − .Then the biorthogonal expansion becomes an orthogonal expansioni=1x = XX T x =n∑−e i (−e T i x) =i=1n∑−e i |e T i x| ∈ R n − (402)i=1and the coordinates of expansion are nonnegative. For this orthant K we haveorthonormality condition X T X = I where X = −I , e i ∈ R n is a standardbasis vector, and −e i is an extreme direction (2.8.1) of K .Of course, this expansion x=XX T x applies more broadly to domain R n ,but then the coordinates each belong to all of R .2.13.8 Biorthogonal expansion, derivationBiorthogonal expansion is a means for determining coordinates in a pointedconic coordinate system characterized by a nonorthogonal basis. Studyof nonorthogonal bases invokes pointed polyhedral cones and their duals;extreme directions of a cone K are assumed to constitute the basis whilethose of the dual cone K ∗ determine coordinates.Unique biorthogonal expansion with respect to K depends upon existenceof its linearly independent extreme directions: Polyhedral cone K must bepointed; then it possesses extreme directions. Those extreme directions mustbe linearly independent to uniquely represent any point in their span.2.71 An orthant is simplicial and selfdual.