v2010.10.26 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 447For any matrix whose range is R(V )= N(1 T ) we get the same result; e.g.,becauser = dim R(XV †TN ) (1033)R(XV ) = {Xz | z ∈ N(1 T )} (1034)and R(V ) = R(V N ) = R(V †TN) (E). These auxiliary matrices (B.4.2) aremore closely related;V = V N V † N(1656)5.7.1.1 Affine dimension r versus rankNow, suppose D is an EDM as defined byD(X) = δ(X T X)1 T + 1δ(X T X) T − 2X T X ∈ EDM N (891)and we premultiply by −V T N and postmultiply by V N . Then because V T N 1=0(898), it is always true that−V T NDV N = 2V T NX T XV N = 2V T N GV N ∈ S N−1 (1035)where G is a Gram matrix.(confer (912))Similarly pre- and postmultiplying by V−V DV = 2V X T XV = 2V GV ∈ S N (1036)always holds because V 1=0 (1646). Likewise, multiplying inner-productform EDM definition (958), it always holds:−V T NDV N = Θ T Θ ∈ S N−1 (962)For any matrix A , rankA T A = rankA = rankA T . [202,0.4] 5.34 So, by(1034), affine dimensionr = rankXV = rankXV N = rankXV †TN= rank Θ= rankV DV = rankV GV = rankVN TDV N = rankVN TGV N(1037)5.34 For A∈R m×n , N(A T A) = N(A). [331,3.3]