v2010.10.26 - Convex Optimization

772 APPENDIX F. NOTATION AND A FEW DEFINITIONS∃∴∀&∝∞≡≈≃∼ =such thatthere existsthereforefor all, or over all(ampersand) andproportional toinfinityequivalent todefined equal to, equal by definitionapproximately equal toisomorphic to or withcongruent to or withHadamard quotient as in, for x,y ∈ R n ,xy [x i/y i , i=1... n ]∈ R n◦ Hadamard product of matrices: x ◦ y [x i y i , i=1... n ]∈ R n⊗Kronecker product of matrices (D.1.2.1)⊕ vector sum of sets X = Y ⊕ Z where every element x∈X hasunique expression x = y + z where y ∈ Y and z ∈ Z ; [307, p.19]then summands are algebraic complements. X = Y ⊕ Z ⇒ X = Y + Z .Now assume Y and Z are nontrivial subspaces. X = Y + Z ⇒X = Y ⊕ Z ⇔ Y ∩ Z =0 [308,1.2] [109,5.8]. Each element froma vector sum (+) of subspaces has unique expression (⊕) when a basisfrom each subspace is linearly independent of bases from all the othersubspaces.⊖likewise, the vector difference of sets