v2004.06.19 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 409By (1267), the product P 1 XP 1 is the one-dimensional orthogonal projectionof X in isomorphic R m2 on the range of vectorized P 1 because, for rankP 1 =1and P 21 =P 1 ∈ S m , (confer (1258))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(1281)The coefficient of orthogonal projection 〈P 1 , X〉= y T Xy/(y T y) is also knownas Rayleigh’s quotient. E.10 When P 1 is rank-one symmetric as in (1280),R(vecP 1 XP 1 ) = R(vec P 1 ) in R m2 (1282)andP 1 XP 1 − X ⊥ P 1 in R m2 (1283)The test for positive semidefiniteness, then, is a test for nonnegativity ofthe coefficient of orthogonal projection of X on the range of each and everyvectorized extreme direction yy T (§2.6.4) from the positive semidefinite conein the ambient space of symmetric matrices.E.10 When y becomes the j th eigenvector s j of diagonalizable X , for example, 〈P 1 , X〉becomes the j th eigenvalue: [58, §III]〈P 1 , X〉| y=sj=s T j( m∑λ i s i wiTi=1s T j s j)s j= λ jSimilarly for y = w j , the j th left-eigenvector,〈P 1 , X〉| y=wj=w T j( m∑λ i s i wiTi=1w T j w 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 [28, §7.1] where (X +X T )/2 isthe symmetric part of X, and s H j denotes the conjugate transpose.