v2009.01.01 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 395
For any matrix whose range is R(V )= N(1 T ) we get the same result; e.g.,
because
r = dim R(XV †T
N ) (938)
R(XV ) = {Xz | z ∈ N(1 T )} (939)
and R(V ) = R(V N ) = R(V †T
N
) (E). These auxiliary matrices (B.4.2) are
more closely related;
V = V N V † N
(1541)
5.7.1.1 Affine dimension r versus rank
Now, suppose D is an EDM as defined by
D(X) ∆ = δ(X T X)1 T + 1δ(X T X) T − 2X T X ∈ EDM N (798)
and we premultiply by −V T N and postmultiply by V N . Then because V T N 1=0
(805), it is always true that
−V T NDV N = 2V T NX T XV N = 2V T N GV N ∈ S N−1 (940)
where G is a Gram matrix.
(confer (820))
Similarly pre- and postmultiplying by V
−V DV = 2V X T XV = 2V GV ∈ S N (941)
always holds because V 1=0 (1531). Likewise, multiplying inner-product
form EDM definition (863), it always holds:
−V T NDV N = Θ T Θ ∈ S N−1 (867)
For any matrix A , rankA T A = rankA = rankA T . [176,0.4] 5.31 So, by
(939), affine dimension
r = rankXV = rankXV N = rankXV †T
N
= rank Θ
= rankV DV = rankV GV = rankVN TDV N = rankVN TGV N
(942)
5.31 For A∈R m×n , N(A T A) = N(A). [287,3.3]