v2010.10.26 - Convex Optimization

128 CHAPTER 2. CONVEX GEOMETRYwhile for one-dimensional matrices, in exception, (M =1,2.7){yy ∈ S | y≠0} = int S + (238)Each and every extreme direction yy T makes the same angle with theidentity matrix in isomorphic R M(M+1)/2 , dependent only on dimension;videlicet, 2.46( )(yy T 〈yy T , I〉 1, I) = arccos = arccos √ ∀y ∈ R M (239)‖yy T ‖ F ‖I‖ F M2.9.2.7.1 Example. Positive semidefinite matrix from extreme directions.Diagonalizability (A.5) of symmetric matrices yields the following results:Any positive semidefinite matrix (1450) in S M can be written in the formA =M∑λ i z i zi T = ÂÂT = ∑ ii=1â i â T i ≽ 0, λ ≽ 0 (240)a conic combination of linearly independent extreme directions (â i â T i or z i z T iwhere ‖z i ‖=1), where λ is a vector of eigenvalues.If we limit consideration to all symmetric positive semidefinite matricesbounded via unity traceC {A ≽ 0 | trA = 1} (91)then any matrix A from that set may be expressed as a convex combinationof linearly independent extreme directions;Implications are:A =M∑λ i z i zi T ∈ C , 1 T λ = 1, λ ≽ 0 (241)i=11. set C is convex (an intersection of PSD cone with hyperplane),2. because the set of eigenvalues corresponding to a given square matrix Ais unique (A.5.0.1), no single eigenvalue can exceed 1 ; id est, I ≽ A3. and the converse holds: set C is an instance of Fantope (91). 2.46 Analogy with respect to the EDM cone is considered in [185, p.162] where it is found:angle is not constant. Extreme directions of the EDM cone can be found in6.4.3.2. Thecone’s axis is −E =11 T − I (1095).