v2010.10.26 - Convex Optimization

612 APPENDIX A. LINEAR ALGEBRAOrigin of the term Schur complement is from complementary inertia:[113,2.4.4] Defineinertia ( G∈ S M) {p,z,n} (1516)where p,z,n respectively represent number of positive, zero, and negativeeigenvalues of G ; id est,M = p + z + n (1517)Then, when A is invertible,and when C is invertible,inertia(G) = inertia(A) + inertia(C − B T A −1 B) (1518)inertia(G) = inertia(C) + inertia(A − BC −1 B T ) (1519)A.4.0.0.1 Example. Equipartition inertia. [55,1.2 exer.17]When A=C = 0, denoting nonincreasingly ordered singular values of matrixB ∈ R m×m by σ(B)∈ R m + , then we have eigenvaluesand[ ] A bb T c([ 0 Bλ(G) = λB T 0])=[σ(B)−Ξσ(B)](1520)inertia(G) = inertia(B T B) + inertia(−B T B) (1521)where Ξ is the order-reversing permutation matrix defined in (1728). A.4.0.0.2 Example. Nonnegative polynomial. [34, p.163]Quadratic multivariate polynomial x T Ax + 2b T x + c is a convex function ofvector x if and only if A ≽ 0, but sublevel set {x | x T Ax + 2b T x + c ≤ 0}is convex if A ≽ 0 yet not vice versa. Schur-form positive semidefinitenessis sufficient for polynomial convexity but necessary and sufficient fornonnegativity:≽ 0 ⇔ [xT 1] [ A bb T c][ x1]≥ 0 ⇔ x T Ax+2b T x+c ≥ 0 (1522)All is extensible to univariate polynomials; e.g., x [t n t n−1 t n−2 · · · t ] T .