v2007.09.17 - Convex Optimization

532 APPENDIX B. SIMPLE MATRICEShas R(V W )= N(1 T ) and orthonormal columns. [4] We defined threeauxiliary V -matrices: V , V N (715), and V W sharing some attributes listedin Table B.4.4. For example, V can be expressedV = V W V T W = V N V † N(1440)but V T W V W= I means V is an orthogonal projector (1680) andV † W = V T W , ‖V W ‖ 2 = 1 , V T W1 = 0 (1441)B.4.4Auxiliary V -matrix TabledimV 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 −1T) V2 −1 IV W N ×(N −1) N −1 N(1 T ) R(1) I V VB.4.5More auxiliary matricesMathar shows [190,2] that any elementary matrix (B.3) of the formV M = I − b1 T ∈ R N×N (1442)such that b T 1 = 1 (confer [112,2]), is an auxiliary V -matrix havingR(V T M ) = N(bT ), R(V M ) = N(1 T )N(V M ) = R(b), N(V T M ) = R(1) (1443)Given X ∈ R n×N , the choice b= 1 N 1 (V M=V ) minimizes ‖X(I − b1 T )‖ F .[114,3.2.1]