v2010.10.26 - Convex Optimization

778 APPENDIX F. NOTATION AND A FEW DEFINITIONSN(A) the subspace: nullspace of A ; N(A) ⊥ R(A T )R nEuclidean n-dimensional real vector space (nonnegative integer n).R 0 = 0, R = R 1 or vector space of unspecified dimension.R m×n[ RmR n ]Euclidean vector space of m by n dimensional real matrices× Cartesian product. R m×n−m R m×(n−m) . K 1 × K 2 =R m × R n = R m+n[K1K 2]C n or C n×nEuclidean complex vector space of respective dimension n and n×nR n + or R n×n+ nonnegative orthant in Euclidean vector space of respective dimensionn and n×nR n −or R n×n−S nnonpositive orthant in Euclidean vector space of respective dimensionn and n×nsubspace of real symmetric n×n matrices; the symmetric matrixsubspace. S = S 1 or symmetric subspace of unspecified dimension.S n⊥ orthogonal complement of S n in R n×n , the antisymmetric matrices (51)S n +int S n +S+(ρ)nEDM N√EDMNPSDconvex cone comprising all (real) symmetric positive semidefinite n×nmatrices, the positive semidefinite coneinterior of convex cone comprising all (real) symmetric positivesemidefinite n×n matrices; id est, positive definite matrices= {X ∈ S n + | rankX ≥ ρ} (260) convex set of all positive semidefiniten×n symmetric matrices whose rank equals or exceeds ρcone of N ×N Euclidean distance matrices in the symmetric hollowsubspacenonconvex cone of N ×N Euclidean absolute distance matrices in thesymmetric hollow subspace (6.3)positive semidefinite