v2010.10.26 - Convex Optimization
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188 CHAPTER 2. CONVEX GEOMETRYthen we find that the pseudoinverse transpose matrixX †T =[Γ 31Γ T 3 Γ 1Γ 41Γ T 4 Γ 2](397)holds the extreme directions of the dual cone. Therefore,x = XX † x (403)is the biorthogonal expansion (395) (E.0.1), and the biorthogonalitycondition (394) can be expressed succinctly (E.1.1) 2.70X † X = I (404)Expansion w=XX † w , for any w ∈ R 2 , is unique if and only if the extremedirections of K are linearly independent; id est, iff X has no nullspace. 2.13.7.1 Pointed cones and biorthogonalityBiorthogonality condition X † X = I from Example 2.13.7.0.1 means Γ 1 andΓ 2 are linearly independent generators of K (B.1.1.1); generators becauseevery x∈ K is their conic combination. From2.10.2 we know that meansΓ 1 and Γ 2 must be extreme directions of K .A biorthogonal expansion is necessarily associated with a pointed closedconvex cone; pointed, otherwise there can be no extreme directions (2.8.1).We will address biorthogonal expansion with respect to a pointed polyhedralcone not full-dimensional in2.13.8.2.13.7.1.1 Example. Expansions implied by diagonalization.(confer6.4.3.2.1) When matrix X ∈ R M×M is diagonalizable (A.5),X = SΛS −1 = [ s 1 · · · s M ] Λ⎣⎡w T1.w T M⎤⎦ =M∑λ i s i wi T (1547)2.70 Possibly confusing is the fact that formula XX † x is simultaneously the orthogonalprojection of x on R(X) (1910), and a sum of nonorthogonal projections of x ∈ R(X) onthe range of each and every column of full-rank X skinny-or-square (E.5.0.0.2).i=1
2.13. DUAL CONE & GENERALIZED INEQUALITY 189coordinates for biorthogonal expansion are its eigenvalues λ i (contained indiagonal matrix Λ) when expanded in S ;⎡X = SS −1 X = [s 1 · · · s M ] ⎣⎤w1 T X. ⎦ =wM TX M∑λ i s i wi T (398)Coordinate value depend upon the geometric relationship of X to its linearlyindependent eigenmatrices s i w T i . (A.5.0.3,B.1.1)Eigenmatrices s i w T i are linearly independent dyads constituted by rightand left eigenvectors of diagonalizable X and are generators of somepointed polyhedral cone K in a subspace of R M×M .When S is real and X belongs to that polyhedral cone K , for example,then coordinates of expansion (the eigenvalues λ i ) must be nonnegative.When X = QΛQ T is symmetric, coordinates for biorthogonal expansionare its eigenvalues when expanded in Q ; id est, for X ∈ S Mi=1X = QQ T X =M∑q i qi T X =i=1M∑λ i q i qi T ∈ S M (399)i=1becomes an orthogonal expansion with orthonormality condition Q T Q=Iwhere λ i is the i th eigenvalue of X , q i is the corresponding i th eigenvectorarranged columnar in orthogonal matrixQ = [q 1 q 2 · · · q M ] ∈ R M×M (400)and where eigenmatrix q i qiT is an extreme direction of some pointedpolyhedral cone K ⊂ S M and an extreme direction of the positive semidefinitecone S M + .Orthogonal expansion is a special case of biorthogonal expansion ofX ∈ aff K occurring when polyhedral cone K is any rotation about theorigin of an orthant belonging to a subspace.Similarly, when X = QΛQ T belongs to the positive semidefinite cone inthe subspace of symmetric matrices, coordinates for orthogonal expansion
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2.13. DUAL CONE & GENERALIZED INEQUALITY 189coordinates for biorthogonal expansion are its eigenvalues λ i (contained indiagonal matrix Λ) when expanded in S ;⎡X = SS −1 X = [s 1 · · · s M ] ⎣⎤w1 T X. ⎦ =wM TX M∑λ i s i wi T (398)Coordinate value depend upon the geometric relationship of X to its linearlyindependent eigenmatrices s i w T i . (A.5.0.3,B.1.1)Eigenmatrices s i w T i are linearly independent dyads constituted by rightand left eigenvectors of diagonalizable X and are generators of somepointed polyhedral cone K in a subspace of R M×M .When S is real and X belongs to that polyhedral cone K , for example,then coordinates of expansion (the eigenvalues λ i ) must be nonnegative.When X = QΛQ T is symmetric, coordinates for biorthogonal expansionare its eigenvalues when expanded in Q ; id est, for X ∈ S Mi=1X = QQ T X =M∑q i qi T X =i=1M∑λ i q i qi T ∈ S M (399)i=1becomes an orthogonal expansion with orthonormality condition Q T Q=Iwhere λ i is the i th eigenvalue of X , q i is the corresponding i th eigenvectorarranged columnar in orthogonal matrixQ = [q 1 q 2 · · · q M ] ∈ R M×M (400)and where eigenmatrix q i qiT is an extreme direction of some pointedpolyhedral cone K ⊂ S M and an extreme direction of the positive semidefinitecone S M + .Orthogonal expansion is a special case of biorthogonal expansion ofX ∈ aff K occurring when polyhedral cone K is any rotation about theorigin of an orthant belonging to a subspace.Similarly, when X = QΛQ T belongs to the positive semidefinite cone inthe subspace of symmetric matrices, coordinates for orthogonal expansion