v2007.09.13 - Convex Optimization

602 APPENDIX E. PROJECTIONis a nonorthogonal projection of matrix X on the range of vectorized dyadP 0 ; from which it follows:P 0 XP 0 = zT Xyz T y〈 〉yz T zyT yzTz T y = z T y , X z T y = 〈P 0 T , X〉 P 0 = 〈P 0 T , X〉〈P0 T , P 0 〉 P 0(1729)Yet this relationship between matrix product and vector inner-product onlyholds for a dyad projector. When nonsymmetric projector P 0 is rank-one asin (1727), therefore,R(vecP 0 XP 0 ) = R(vec P 0 ) in R m2 (1730)andP 0 XP 0 − X ⊥ P T 0 in R m2 (1731)E.6.2.1.1 Example. λ as coefficients of nonorthogonal projection.Any diagonalization (A.5)X = SΛS −1 =m∑λ i s i wi T ∈ R m×m (1334)i=1may be expressed as a sum of one-dimensional nonorthogonal projectionsof X , each on the range of a vectorized eigenmatrix P j ∆ = s j w T j ;∑X = m 〈(Se i e T jS −1 ) T , X〉 Se i e T jS −1i,j=1∑= m ∑〈(s j wj T ) T , X〉 s j wj T + m 〈(Se i e T jS −1 ) T , SΛS −1 〉 Se i e T jS −1j=1∑= m 〈(s j wj T ) T , X〉 s j wjTj=1i,j=1j ≠ i∆ ∑= m ∑〈Pj T , X〉 P j = m s j wj T Xs j wjTj=1∑= m λ j s j wjTj=1j=1∑= m P j XP jj=1(1732)This biorthogonal expansion of matrix X is a sum of nonorthogonalprojections because the term outside the projection coefficient 〈 〉 is not