v2009.01.01 - Convex Optimization

590 APPENDIX B. SIMPLE MATRICES
has R(V W )= N(1 T ) and orthonormal columns. [5] We defined three
auxiliary V -matrices: V , V N (804), and V W sharing some attributes listed
in Table B.4.4. For example, V can be expressed
V = V W V T W = V N V † N
(1541)
but V T W V W
= I means V is an orthogonal projector (1788) and
V † W = V T W , ‖V W ‖ 2 = 1 , V T W1 = 0 (1542)
B.4.4
Auxiliary V -matrix Table
dimV rankV R(V ) N(V T ) V T V V V T V V †
V N ×N N −1 N(1 T ) R(1) V V V
[ ]
V N N ×(N −1) N −1 N(1 T 1
) R(1) (I + 2 11T 1 N −1 −1
T
) V
2 −1 I
V W N ×(N −1) N −1 N(1 T ) R(1) I V V
B.4.5
More auxiliary matrices
Mathar shows [223,2] that any elementary matrix (B.3) of the form
V M = I − b1 T ∈ R N×N (1543)
such that b T 1 = 1 (confer [136,2]), is an auxiliary V -matrix having
R(V T M ) = N(bT ), R(V M ) = N(1 T )
N(V M ) = R(b), N(V T M ) = R(1) (1544)
Given X ∈ R n×N , the choice b= 1 N 1 (V M=V ) minimizes ‖X(I − b1 T )‖ F .
[138,3.2.1]