v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
610 APPENDIX E. PROJECTIONPerpendicularity (1757) 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 (1760)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 )}(1761)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 (1757) 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 (1762)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 }(1763)
E.7. ON VECTORIZED MATRICES OF HIGHER RANK 611characterized by the doublet u1 T + 1u T (B.2). E.12 Defining thegeometric center mapping V(X) = −V XV 1 consistently with (801), then2N(V)= R(I − V) on domain S n analogously to vector projectors (E.2);id est,N(V) = S n⊥c (1764)a subspace of S n whose dimension is dim S n⊥c = n in isomorphic R n(n+1)/2 .Intuitively, operator V is an orthogonal projector; any argumentduplicitously in its range is a fixed point. So, this symmetric operator’snullspace must be orthogonal to its range.Now compare the subspace of symmetric matrices having all zeros in thefirst row and columnS n 1∆= {Y ∈ S n | Y e 1 = 0}{[ ] [ ] }0 0T 0 0T= X | X ∈ S n0 I 0 I{ [0 √ ] T [ √ ]= 2VN Z 0 2VN | Z ∈ SN}(1765)[ ] 0 0Twhere P = is an orthogonal projector. Then, similarly, PXP is0 Ithe orthogonal projection of any X ∈ S n on S n 1 in the Euclidean sense (1757),andS n⊥1{[ ] [ ] }0 0T 0 0TX − X | X ∈ S n ⊂ S n0 I 0 I= { (1766)ue T 1 + e 1 u T | u∈ R n}∆=Obviously, S n 1 ⊕ S n⊥1 = S n . E.12 Proof.{X − V X V | X ∈ S n } = {X − (I − 1 n 11T )X(I − 11 T 1 n ) | X ∈ Sn }Because {X1 | X ∈ S n } = R n ,= { 1 n 11T X + X11 T 1 n − 1 n 11T X11 T 1 n | X ∈ Sn }{X − V X V | X ∈ S n } = {1ζ T + ζ1 T − 11 T (1 T ζ 1 n ) | ζ ∈Rn }= {1ζ T (I − 11 T 12n ) + (I − 12n 11T )ζ1 T | ζ ∈R n }where I − 12n 11T is invertible.
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610 APPENDIX E. PROJECTIONPerpendicularity (1757) 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 (1760)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 )}(1761)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 (1757) 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 (1762)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 }(1763)