v2007.09.13 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 603identical 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 (1732),andR(vecP j XP j ) = R(vec s j w T j ) = R(vec P j ) in R m2 (1733)P j XP j − X ⊥ P T j in R m2 (1734)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; [11,3.3] [247,3.2][273,2.2] [285,1-8]〈y,x〉〈y,y〉 y = yT xy T y y = yyTy T y x = ∆ P 1 x (1735)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 ) (1736)Orthogonal projection on a vectorized matrixFrom (1735), 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 (1737)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.