v2007.09.13 - Convex Optimization

604 APPENDIX E. PROJECTIONE.6.4.1Orthogonal projection on dyadThere is opportunity for insight when Y is a dyad yz T (B.1): Instead givenX ∈ R m×n , y ∈ R m , and z ∈ R n〈yz T , X〉〈yz T , yz T 〉 yzT = yT Xzy T y z T z yzT (1738)is the one-dimensional orthogonal projection of X in isomorphic R mn onthe range of vectorized yz T . To reveal the obscured symmetric projectionmatrices P 1 and P 2 we rewrite (1738):y T Xzy T y z T z yzT =yyTy T y X zzTz T z∆= P 1 XP 2 (1739)So for projector dyads, projection (1739) is the orthogonal projection in R mnif and only if projectors P 1 and P 2 are symmetric; E.9 in other words,andfor orthogonal projection on the range of a vectorized dyad yz T , theterm outside the vector inner-products 〈 〉 in (1738) must be identicalto the terms inside in three places.When P 1 and P 2 are rank-one symmetric projectors as in (1739), (30)When y=z then P 1 =P 2 =P T 2 andR(vecP 1 XP 2 ) = R(vec yz T ) in R mn (1740)P 1 XP 2 − X ⊥ yz T in R mn (1741)P 1 XP 1 = 〈P 1 , X〉 P 1 = 〈P 1 , X〉〈P 1 , P 1 〉 P 1 (1742)E.9 For diagonalizable X ∈ R m×m (A.5), its orthogonal projection in isomorphic R m2 onthe range of vectorized yz T ∈ R m×m becomes:P 1 XP 2 =m∑λ i P 1 s i wi T P 2i=1When R(P 1 ) = R(w j ) and R(P 2 ) = R(s j ), the j th dyad term from the diagonalizationis isolated but only, in general, to within a scale factor because neither set of left orright eigenvectors is necessarily orthonormal unless X is normal [298,3.2]. Yet whenR(P 2 )= R(s k ) , k≠j ∈{1... m}, then P 1 XP 2 =0.