v2010.10.26 - Convex Optimization

634 APPENDIX B. SIMPLE MATRICESB.1 Rank-one matrix (dyad)Any matrix formed from the unsigned outer product of two vectors,Ψ = uv T ∈ R M×N (1606)where u ∈ R M and v ∈ R N , is rank-one and called a dyad. Conversely, anyrank-one matrix must have the form Ψ . [202, prob.1.4.1] Product −uv T isa negative dyad. For matrix products AB T , in general, we haveR(AB T ) ⊆ R(A) , N(AB T ) ⊇ N(B T ) (1607)with equality when B =A [331,3.3,3.6] B.1 or respectively when B isinvertible and N(A)=0. Yet for all nonzero dyads we haveR(uv T ) = R(u) , N(uv T ) = N(v T ) ≡ v ⊥ (1608)where dim v ⊥ =N −1.It is obvious a dyad can be 0 only when u or v is 0 ;Ψ = uv T = 0 ⇔ u = 0 or v = 0 (1609)The matrix 2-norm for Ψ is equivalent to Frobenius’ norm;‖Ψ‖ 2 = σ 1 = ‖uv T ‖ F = ‖uv T ‖ 2 = ‖u‖ ‖v‖ (1610)When u and v are normalized, the pseudoinverse is the transposed dyad.Otherwise,Ψ † = (uv T ) † vu T=(1611)‖u‖ 2 ‖v‖ 2B.1 Proof. R(AA T ) ⊆ R(A) is obvious.R(AA T ) = {AA T y | y ∈ R m }⊇ {AA T y | A T y ∈ R(A T )} = R(A) by (142)