v2010.10.26 - Convex Optimization

728 APPENDIX E. PROJECTION(A.2) It is a fact that y T Xy is always proportional to a coefficient oforthogonal projection; letting z in formula (1975) become y ∈ R m , thenP 2 =P 1 =yy T /y T y=yy T /‖yy T ‖ 2 (confer (1610)) and formula (1976) becomes〈yy T , X〉〈yy T , yy T 〉 yyT = yT Xyy T yyy Ty T y = yyTy T y X yyTy T y P 1XP 1 (1987)By (1974), product P 1 XP 1 is the one-dimensional orthogonal projection ofX in isomorphic R m2 on the range of vectorized P 1 because, for rankP 1 =1and P 21 =P 1 ∈ S m (confer (1966))P 1 XP 1 = yT Xyy T y〈 〉yy T yyT yyTy T y = y T y , X y T y = 〈P 1 , X〉 P 1 = 〈P 1 , X〉〈P 1 , P 1 〉 P 1(1988)The coefficient of orthogonal projection 〈P 1 , X〉= y T Xy/(y T y) is also knownas Rayleigh’s quotient. E.12 When P 1 is rank-one symmetric as in (1987),R(vec P 1 XP 1 ) = R(vec P 1 ) in R m2 (1989)andP 1 XP 1 − X ⊥ P 1 in R m2 (1990)E.12 When y becomes the j th eigenvector s j of diagonalizable X , for example, 〈P 1 , X 〉becomes the j th eigenvalue: [195,III]( m∑)s T j λ i s i wiT s ji=1〈P 1 , X 〉| y=sj=s T j s = λ jjSimilarly for y = w j , the j th left-eigenvector,( m∑〈P 1 , X 〉| y=wj=w T jλ i s i wiTi=1wj Tw j)w j= λ jA quandary may arise regarding the potential annihilation of the antisymmetric part ofX when s T j Xs j is formed. Were annihilation to occur, it would imply the eigenvalue thusfound came instead from the symmetric part of X . The quandary is resolved recognizingthat diagonalization of real X admits complex eigenvectors; hence, annihilation could onlycome about by forming re(s H j Xs j) = s H j (X +XT )s j /2 [202,7.1] where (X +X T )/2 isthe symmetric part of X , and s H j denotes conjugate transpose.