v2009.01.01 - Convex Optimization

A.3. PROPER STATEMENTS 547
A.3.1
Semidefiniteness, eigenvalues, nonsymmetric
When A∈ R n×n , let λ ( 1
2 (A +AT ) ) ∈ R n denote eigenvalues of the
symmetrized matrix A.8 arranged in nonincreasing order.
By positive semidefiniteness of A∈ R n×n we mean, A.9 [235,1.3.1]
(conferA.3.1.0.1)
x T Ax ≥ 0 ∀x∈ R n ⇔ A +A T ≽ 0 ⇔ λ(A +A T ) ≽ 0 (1347)
(2.9.0.1)
A ≽ 0 ⇒ A T = A (1348)
A ≽ B ⇔ A −B ≽ 0 A ≽ 0 or B ≽ 0 (1349)
x T Ax≥0 ∀x A T = A (1350)
Matrix symmetry is not intrinsic to positive semidefiniteness;
A T = A, λ(A) ≽ 0 ⇒ x T Ax ≥ 0 ∀x (1351)
If A T = A thenλ(A) ≽ 0 ⇐ A T = A, x T Ax ≥ 0 ∀x (1352)
λ(A) ≽ 0 ⇔ A ≽ 0 (1353)
meaning, matrix A belongs to the positive semidefinite cone in the
subspace of symmetric matrices if and only if its eigenvalues belong to
the nonnegative orthant.
〈A , A〉 = 〈λ(A), λ(A)〉 (38)
For µ∈ R , A∈ R n×n , and vector λ(A)∈ C n holding the ordered
eigenvalues of A
λ(µI + A) = µ1 + λ(A) (1354)
Proof: A=MJM −1 and µI + A = M(µI + J)M −1 where J is
the Jordan form for A ; [287,5.6, App.B] id est, δ(J) = λ(A) , so
λ(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 ( 1
2 (A +AT ) ) = λ(A +A T )/2.
A.9 Strang agrees [287, p.334] it is not λ(A) that requires observation. Yet he is mistaken
by proposing the Hermitian part alone x H (A+A H )x be tested, because the anti-Hermitian
part does not vanish under complex test unless A is Hermitian. (1338)