v2007.09.13 - Convex Optimization

678 APPENDIX G. NOTATION AND A FEW DEFINITIONSXGrnNdomonontoepispanpoint list (having cardinality N) arranged columnar in R n×N , or set ofgenerators, or extreme directions, or matrix variableGram matrix X T Xaffine dimensionEuclidean dimension of list X , or integercardinality of list X , or integerfunction domainfunction f(x) on A means A is domf , or projection of x on Ameans A is Euclidean body on which projection of x is madefunction f(x) maps onto M means f over its domain is a surjectionwith respect to Mfunction epigraphas in spanA = R(A) = {Ax | x∈ R n } when A is a matrixR(A) the subspace: range of A , or span basis R(A) ; R(A) ⊥ N(A T )basis R(A)columnar basis for range of A , or a minimal set constituting generatorsfor the vertex-description of R(A) , or a linearly independent set ofvectors spanning R(A)N(A) the subspace: nullspace of A ; N(A) ⊥ R(A T )× Cartesian product[ ] RmR n R m × R n = R m+nR m×nEuclidean vector space of m by n dimensional real matricesR n Euclidean n-dimensional real vector space (nonnegative integer n)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×n