v2007.09.13 - Convex Optimization

674 APPENDIX G. NOTATION AND A FEW DEFINITIONS⊗⊕⊖⊞Kronecker product of matrices (D.1.2.1)vector sum of sets X = Y ⊕ Z where every element x∈X has uniqueexpression x = y + z where y ∈ Y and z ∈ Z ; [228, p.19] then thesummands 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 [229,1.2] [73,5.8]. Each element froma vector sum (+) of subspaces has a unique representation (⊕) when abasis from each subspace is linearly independent of bases from all theother subspaces.likewise, the vector difference of setsorthogonal vector sum of sets X = Y ⊞ Z where every element x∈Xhas unique orthogonal expression x = y + z where y ∈ Y , z ∈ Z ,and y ⊥ z . [245, p.51] X = Y ⊞ Z ⇒ X = Y + Z . If Z ⊆ Y ⊥ thenX = Y ⊞ Z ⇔ X = Y ⊕ Z . [73,5.8] If Z = Y ⊥ then the summandsare orthogonal complements.± plus or minus⊥as in A ⊥ B meaning A is orthogonal to B (and vice versa), whereA and B are sets, vectors, or matrices. When A and B arevectors (or matrices under Frobenius norm), A ⊥ B ⇔ 〈A,B〉 = 0⇔ ‖A + B‖ 2 = ‖A‖ 2 + ‖B‖ 2\ as in \A means logical not A , or relative complement of set A ;id est, \A = {x /∈A} ; e.g., B\A ∆ = {x∈ B | x /∈A} ≡ B ∩\A⇒ or ⇐ sufficiency or necessity, implies; e.g., A ⇒ B ⇔ \A ⇐ \B⇔is or ←→if and only if (iff) or corresponds to or necessary and sufficient orthe same asas in A is B means A ⇒ B ; conventional usage of English languageby mathematiciansdoes not implyis replaced with; substitution, assignmentgoes to, or approaches, or maps to