v2010.10.26 - Convex Optimization

B.1. RANK-ONE MATRIX (DYAD) 637B.1.1Dyad independenceNow we consider a sum of dyads like (1606) as encountered in diagonalizationand singular value decomposition:( k∑)k∑R s i wiT = R ( )k∑s i wiT = R(s i ) ⇐ w i ∀i are l.i. (1621)i=1i=1range of summation is the vector sum of ranges. B.3 (Theorem B.1.1.1.1)Under the assumption the dyads are linearly independent (l.i.), then vectorsums are unique (p.772): for {w i } l.i. and {s i } l.i.( k∑R s i wiTi=1)i=1= R ( s 1 w T 1)⊕ ... ⊕ R(sk w T k)= R(s1 ) ⊕ ... ⊕ R(s k ) (1622)B.1.1.0.1 Definition. Linearly independent dyads. [211, p.29 thm.11][339, p.2] The set of k dyads{si w T i | i=1... k } (1623)where s i ∈ C M and w i ∈ C N , is said to be linearly independent iff()k∑rank SW T s i wiT = k (1624)i=1where S [s 1 · · · s k ] ∈ C M×k and W [w 1 · · · w k ] ∈ C N×k .△Dyad independence does not preclude existence of a nullspace N(SW T ) ,as defined, nor does it imply SW T were full-rank. In absence of assumptionof independence, generally, rankSW T ≤ k . Conversely, any rank-k matrixcan be written in the form SW T by singular value decomposition. (A.6)B.1.1.0.2 Theorem. Linearly independent (l.i.) dyads.Vectors {s i ∈ C M , i=1... k} are l.i. and vectors {w i ∈ C N , i=1... k} arel.i. if and only if dyads {s i wi T ∈ C M×N , i=1... k} are l.i.⋄B.3 Move of range R to inside summation depends on linear independence of {w i }.