v2007.09.13 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 601E.6 Vectorization interpretation,projection on a matrixE.6.1Nonorthogonal projection on a vectorNonorthogonal projection of vector x on the range of vector y isaccomplished using a normalized dyad P 0 (B.1); videlicet,〈z,x〉〈z,y〉 y = zT xz T y y = yzTz T y x ∆ = P 0 x (1724)where 〈z,x〉/〈z,y〉 is the coefficient of projection on y . Because P0 2 =P 0and R(P 0 )= R(y) , rank-one matrix P 0 is a nonorthogonal projectorprojecting on R(y) . The direction of nonorthogonal projection is orthogonalto z ; id est,P 0 x − x ⊥ R(P0 T ) (1725)E.6.2Nonorthogonal projection on vectorized matrixFormula (1724) is extensible. Given X,Y,Z ∈ R m×n , we have theone-dimensional nonorthogonal projection of X in isomorphic R mn on therange of vectorized Y : (2.2)〈Z , X〉Y , 〈Z , Y 〉 ≠ 0 (1726)〈Z , Y 〉where 〈Z , X〉/〈Z , Y 〉 is the coefficient of projection. The inequalityaccounts for the fact: projection on R(vec Y ) is in a direction orthogonal tovec Z .E.6.2.1Nonorthogonal projection on dyadNow suppose we have nonorthogonal projector dyadAnalogous to (1724), for X ∈ R m×mP 0 = yzTz T y ∈ Rm×m (1727)P 0 XP 0 =yzTz T y X yzTz T y = zT Xy(z T y) 2 yzT = 〈zyT , X〉〈zy T , yz T 〉 yzT (1728)