v2006.03.09 - Convex Optimization

438 APPENDIX A. LINEAR ALGEBRAFor A,B ∈ S n [261,6.2]A ≽ B ≽ 0 ⇒ A 1/2 ≽ B 1/2 (1093)For A,B ∈ S n and AB = BA [261,6.2, prob.3]A ≽ B ≽ 0 ⇒ A k ≽ B k , k=1, 2,... (1094)A.3.1.0.1 Theorem. Positive semidefinite ordering of eigenvalues.For A,B∈ R M×M , place the eigenvalues of each symmetrized matrix intothe respective vectors λ ( 1(A 2 +AT ) ) , λ ( 1(B 2 +BT ) ) ∈ R M . Then, [212,6]x T Ax ≥ 0 ∀x ⇔ λ ( A +A T) ≽ 0 (1095)x T Ax > 0 ∀x ≠ 0 ⇔ λ ( A +A T) ≻ 0 (1096)because x T (A −A T )x=0. (1039) Now arrange the entries of λ ( 1(A 2 +AT ) )and λ ( 1(B 2 +BT ) ) in nonincreasing order so λ ( 1(A 2 +AT ) ) holds the1largest eigenvalue of symmetrized A while λ ( 1(B 2 +BT ) ) holds the largest1eigenvalue of symmetrized B , and so on. Then [125,7.7, prob.1, prob.9]for κ ∈ Rx T Ax ≥ x T Bx ∀x ⇒ λ ( A +A T) ≽ λ ( B +B T)x T Ax ≥ x T Ixκ ∀x ⇔ λ ( 12 (A +AT ) ) ≽ κ1(1097)Now let A,B ∈ S M have diagonalizations A=QΛQ T and B =UΥU T withλ(A)=δ(Λ) and λ(B)=δ(Υ) arranged in nonincreasing order. ThenA ≽ B ⇔ λ(A−B) ≽ 0 (1098)A ≽ B ⇒ λ(A) ≽ λ(B) (1099)A ≽ B λ(A) ≽ λ(B) (1100)S T AS ≽ B ⇐ λ(A) ≽ λ(B) (1101)where S = QU T . [261,7.5]⋄