v2007.09.17 - Convex Optimization

490 APPENDIX A. LINEAR ALGEBRAA.3.1Semidefiniteness, eigenvalues, nonsymmetricWhen A∈ R n×n , let λ ( 12 (A +AT ) ) ∈ R n denote eigenvalues of thesymmetrized matrix A.8 arranged in nonincreasing order.By positive semidefiniteness of A∈ R n×n we mean, A.9 [200,1.3.1](conferA.3.1.0.1)x T Ax ≥ 0 ∀x∈ R n ⇔ A +A T ≽ 0 ⇔ λ(A +A T ) ≽ 0 (1247)(2.9.0.1)A ≽ 0 ⇒ A T = A (1248)A ≽ B ⇔ A −B ≽ 0 A ≽ 0 or B ≽ 0 (1249)x T Ax≥0 ∀x A T = A (1250)Matrix symmetry is not intrinsic to positive semidefiniteness;A T = A, λ(A) ≽ 0 ⇒ x T Ax ≥ 0 ∀x (1251)If A T = A thenλ(A) ≽ 0 ⇐ A T = A, x T Ax ≥ 0 ∀x (1252)λ(A) ≽ 0 ⇔ A ≽ 0 (1253)meaning, matrix A belongs to the positive semidefinite cone in thesubspace of symmetric matrices if and only if its eigenvalues belong tothe nonnegative orthant.〈A , A〉 = 〈λ(A), λ(A)〉 (1254)For µ∈ R , A∈ R n×n , and vector λ(A)∈ C n holding the orderedeigenvalues of Aλ(µI + A) = µ1 + λ(A) (1255)Proof: A=MJM −1 and µI + A = M(µI + J)M −1 where J isthe Jordan form for A ; [249,5.6, App.B] id est, δ(J) = λ(A) , soλ(µI + A) = δ(µI + J) because µI + J is also a Jordan form. A.8 The symmetrization of A is (A +A T )/2. λ ( 12 (A +AT ) ) = λ(A +A T )/2.A.9 Strang agrees [249, p.334] it is not λ(A) that requires observation. Yet he is mistakenby proposing the Hermitian part alone x H (A+A H )x be tested, because the anti-Hermitianpart does not vanish under complex test unless A is Hermitian. (1238)