v2010.10.26 - Convex Optimization

726 APPENDIX E. PROJECTIONmeaning, P 1 XP 1 is equivalent to orthogonal projection of matrix X onthe range of vectorized projector dyad P 1 . Yet this relationship betweenmatrix product and vector inner-product does not hold for general symmetricprojector matrices.E.6.4.1.1 Example. Eigenvalues λ as coefficients of orthogonal projection.Let C represent any convex subset of subspace S M , and let C 1 be any elementof C . Then C 1 can be expressed as the orthogonal expansionC 1 =M∑i=1M∑〈E ij , C 1 〉 E ij ∈ C ⊂ S M (1980)j=1j ≥ iwhere E ij ∈ S M is a member of the standard orthonormal basis for S M(59). This expansion is a sum of one-dimensional orthogonal projectionsof C 1 ; each projection on the range of a vectorized standard basis matrix.Vector inner-product 〈E ij , C 1 〉 is the coefficient of projection of svec C 1 onR(svec E ij ).When C 1 is any member of a convex set C whose dimension is L ,Carathéodory’s theorem [113] [307] [199] [41] [42] guarantees that no morethan L +1 affinely independent members from C are required to faithfullyrepresent C 1 by their linear combination. E.11Dimension of S M is L=M(M+1)/2 in isometrically isomorphicR M(M+1)/2 . Yet because any symmetric matrix can be diagonalized, (A.5.1)C 1 ∈ S M is a linear combination of its M eigenmatrices q i q T i (A.5.0.3)weighted by its eigenvalues λ i ;C 1 = QΛQ T =M∑λ i q i qi T (1981)i=1where Λ ∈ S M is a diagonal matrix having δ(Λ) i =λ i , and Q=[q 1 · · · q M ]is an orthogonal matrix in R M×M containing corresponding eigenvectors.To derive eigenvalue decomposition (1981) from expansion (1980), Mstandard basis matrices E ij are rotated (B.5) into alignment with the ME.11 Carathéodory’s theorem guarantees existence of a biorthogonal expansion for anyelement in aff C when C is any pointed closed convex cone.