v2007.11.26 - Convex Optimization

626 APPENDIX E. PROJECTIONPerpendicularity (1788) establishes uniqueness [73,4.9] of projection P 1 XP 2on a matrix subspace. The minimum-distance projector is the orthogonalprojector, and vice versa.E.7.2.0.2 Example. PXP redux & N(V).Suppose we define a subspace of m ×n matrices, each elemental matrixhaving columns constituting a list whose geometric center (5.5.1.0.1) is theorigin in R m :R m×nc∆= {Y ∈ R m×n | Y 1 = 0}= {Y ∈ R m×n | N(Y ) ⊇ 1} = {Y ∈ R m×n | R(Y T ) ⊆ N(1 T )}= {XV | X ∈ R m×n } ⊂ R m×n (1791)the nonsymmetric geometric center subspace. Further suppose V ∈ S n isa projection matrix having N(V )= R(1) and R(V ) = N(1 T ). Then linearmapping T(X)=XV is the orthogonal projection of any X ∈ R m×n on R m×ncin the Euclidean (vectorization) sense because V is symmetric, N(XV )⊇1,and R(VX T )⊆ N(1 T ).Now suppose we define a subspace of symmetric n ×n matrices each ofwhose columns constitute a list having the origin in R n as geometric center,S n c∆= {Y ∈ S n | Y 1 = 0}= {Y ∈ S n | N(Y ) ⊇ 1} = {Y ∈ S n | R(Y ) ⊆ N(1 T )}(1792)the geometric center subspace. Further suppose V ∈ S n is a projectionmatrix, the same as before. Then V XV is the orthogonal projection ofany X ∈ S n on S n c in the Euclidean sense (1788) because V is symmetric,V XV 1=0, and R(V XV )⊆ N(1 T ). Two-sided projection is necessary onlyto remain in the ambient symmetric matrix subspace. ThenS n c = {V XV | X ∈ S n } ⊂ S n (1793)has dim S n c = n(n−1)/2 in isomorphic R n(n+1)/2 . We find its orthogonalcomplement as the aggregate of all negative directions of orthogonalprojection on S n c : the translation-invariant subspace (5.5.1.1)S n⊥c∆= {X − V XV | X ∈ S n } ⊂ S n= {u1 T + 1u T | u∈ R n }(1794)