v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
724 APPENDIX E. PROJECTIONidentical to the inside-term. (E.6.4) The eigenvalues λ j are coefficients ofnonorthogonal projection of X , while the remaining M(M −1)/2 coefficients(for i≠j) are zeroed by projection. When P j is rank-one as in (1969),andR(vecP j XP j ) = R(vecs j w T j ) = R(vecP j ) in R m2 (1970)P j XP j − X ⊥ P T j in R m2 (1971)Were matrix X symmetric, then its eigenmatrices would also be. So theone-dimensional projections would become orthogonal. (E.6.4.1.1) E.6.3Orthogonal projection on a vectorThe formula for orthogonal projection of vector x on the range of vector y(one-dimensional projection) is basic analytic geometry; [13,3.3] [331,3.2][365,2.2] [380,1-8]〈y,x〉〈y,y〉 y = yT xy T y y = yyTy T y x P 1x (1972)where 〈y,x〉/〈y,y〉 is the coefficient of projection on R(y). An equivalentdescription is: Vector P 1 x is the orthogonal projection of vector x onR(P 1 )= R(y). Rank-one matrix P 1 is a projection matrix because P 2 1 =P 1 .The direction of projection is orthogonalbecause P T 1 = P 1 .E.6.4P 1 x − x ⊥ R(P 1 ) (1973)Orthogonal projection on a vectorized matrixFrom (1972), given instead X, Y ∈ R m×n , we have the one-dimensionalorthogonal projection of matrix X in isomorphic R mn on the range ofvectorized Y : (2.2)〈Y , X〉〈Y , Y 〉 Y (1974)where 〈Y , X〉/〈Y , Y 〉 is the coefficient of projection.For orthogonal projection, the term outside the vector inner-products 〈 〉must be identical to the terms inside in three places.
E.6. VECTORIZATION INTERPRETATION, 725E.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 (1975)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 (1975):y T Xzy T y z T z yzT =yyTy T y X zzTz T z P 1 XP 2 (1976)So for projector dyads, projection (1976) is the orthogonal projection in R mnif and only if projectors P 1 and P 2 are symmetric; E.10 in other words,andfor orthogonal projection on the range of a vectorized dyad yz T , theterm outside the vector inner-products 〈 〉 in (1975) must be identicalto the terms inside in three places.When P 1 and P 2 are rank-one symmetric projectors as in (1976), (37)When y=z then P 1 =P 2 =P T 2 andR(vec P 1 XP 2 ) = R(vecyz T ) in R mn (1977)P 1 XP 2 − X ⊥ yz T in R mn (1978)P 1 XP 1 = 〈P 1 , X〉 P 1 = 〈P 1 , X〉〈P 1 , P 1 〉 P 1 (1979)E.10 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 or righteigenvectors is necessarily orthonormal unless X is a normal matrix [393,3.2]. Yet whenR(P 2 )= R(s k ) , k≠j ∈{1... m}, then P 1 XP 2 =0.
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E.6. VECTORIZATION INTERPRETATION, 725E.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 (1975)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 (1975):y T Xzy T y z T z yzT =yyTy T y X zzTz T z P 1 XP 2 (1976)So for projector dyads, projection (1976) is the orthogonal projection in R mnif and only if projectors P 1 and P 2 are symmetric; E.10 in other words,andfor orthogonal projection on the range of a vectorized dyad yz T , theterm outside the vector inner-products 〈 〉 in (1975) must be identicalto the terms inside in three places.When P 1 and P 2 are rank-one symmetric projectors as in (1976), (37)When y=z then P 1 =P 2 =P T 2 andR(vec P 1 XP 2 ) = R(vecyz T ) in R mn (1977)P 1 XP 2 − X ⊥ yz T in R mn (1978)P 1 XP 1 = 〈P 1 , X〉 P 1 = 〈P 1 , X〉〈P 1 , P 1 〉 P 1 (1979)E.10 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 or righteigenvectors is necessarily orthonormal unless X is a normal matrix [393,3.2]. Yet whenR(P 2 )= R(s k ) , k≠j ∈{1... m}, then P 1 XP 2 =0.