v2009.01.01 - Convex Optimization

558 APPENDIX A. LINEAR ALGEBRA
Origin of the term Schur complement is from complementary inertia:
[96,2.4.4] Define
inertia ( G∈ S M) ∆ = {p,z,n} (1415)
where p,z,n respectively represent number of positive, zero, and negative
eigenvalues of G ; id est,
M = p + z + n (1416)
Then, when A is invertible,
and when C is invertible,
inertia(G) = inertia(A) + inertia(C − B T A −1 B) (1417)
inertia(G) = inertia(C) + inertia(A − BC −1 B T ) (1418)
When A=C =0, denoting by σ(B)∈ R m + the nonincreasingly ordered
singular values of matrix B ∈ R m×m , then we have the eigenvalues
[48,1.2, prob.17]
([ 0 B
λ(G) = λ
B T 0
])
=
[
σ(B)
−Ξσ(B)
]
(1419)
and
inertia(G) = inertia(B T B) + inertia(−B T B) (1420)
where Ξ is the order-reversing permutation matrix defined in (1608).
A.4.0.0.1 Example. Nonnegative polynomial. [31, p.163]
Schur-form positive semidefiniteness is sufficient for quadratic polynomial
convexity in x , but it is necessary and sufficient for nonnegativity; videlicet,
for all compatible x
[x T 1] [ A b
b T c
] [ x
1
]
≥ 0 ⇔ x T Ax + 2b T x + c ≥ 0 (1421)
Sublevel set {x | x T Ax + 2b T x + c ≤ 0} is convex if A ≽ 0, but the quadratic
polynomial is a convex function if and only if A ≽ 0.