v2007.09.13 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 337For any matrix whose range is R(V )= N(1 T ) we get the same result; e.g.,becauser = dim R(XV †TN ) (838)R(XV ) = {Xz | z ∈ N(1 T )} (839)and R(V ) = R(V N ) = R(V †TN) (E). These auxiliary matrices (B.4.2) aremore closely related;V = V N V † N(1435)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 (705)and we premultiply by −V T N and postmultiply by V N . Then because V T N 1=0(712), it is always true that−V T NDV N = 2V T NX T XV N = 2V T N GV N ∈ S N−1 (840)where G is a Gram matrix.(confer (727))Similarly pre- and postmultiplying by V−V DV = 2V X T XV = 2V GV ∈ S N (841)always holds because V 1=0 (1425). Likewise, multiplying inner-productform EDM definition (770), it always holds:−V T NDV N = Θ T Θ ∈ S N−1 (774)For any matrix A , rankA T A = rankA = rankA T . [149,0.4] 5.30 So, by(839), affine dimensionr = rankXV = rankXV N = rankXV †TN= rank Θ= rankV DV = rankV GV = rankVN TDV N = rankVN TGV N(842)5.30 For A∈R m×n , N(A T A) = N(A). [247,3.3]