v2010.10.26 - Convex Optimization

646 APPENDIX B. SIMPLE MATRICESB.4.3The skinny matrix⎡V W ⎢⎣Orthonormal auxiliary matrix V W−1 √N1 + −1N+ √ N−1N+ √ N.−1N+ √ N√−1−1N· · · √N−1N+ √ N......−1N+ √ N· · ·−1N+ √ N...−1N+ √ N...· · · 1 + −1N+ √ N.⎤∈ R N×N−1 (1655)⎥⎦has R(V W )= N(1 T ) and orthonormal columns. [6] We defined threeauxiliary V -matrices: V , V N (897), 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(1656)but VW TV W = I means V is an orthogonal projector (1913) andB.4.4V † W = V T W , ‖V W ‖ 2 = 1 , V T W1 = 0 (1657)Auxiliary 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 [259,2] that any elementary matrix (B.3) of the formV S = I − b1 T ∈ R N×N (1658)such that b T 1 = 1 (confer [162,2]), is an auxiliary V -matrix havingR(VS T)= N(bT ), R(V S ) = N(1 T )N(V S ) = R(b), N(VS T)= R(1) (1659)