v2006.03.09 - Convex Optimization

A.4. SCHUR COMPLEMENT 443A.4.0.0.1 Example. Sparse Schur conditions.Setting matrix A to the identity simplifies the Schur conditions.consequence relates the definiteness of three quantities:[ ] I B≽ 0 ⇔CB TC − B T B ≽ 0 ⇔[ I 00 T C −B T BOne]≽ 0 (1113)Origin of the term Schur complement is from complementary inertia:[61,2.4.4] Defineinertia ( G∈ S M) ∆ = {p,z,n} (1114)where p,z,n respectively represent the number of positive, zero, andnegative eigenvalues of G ; id est,Then, when C is invertible,and when A is invertible,M = p + z + n (1115)inertia(G) = inertia(C) + inertia(A − BC −1 B T ) (1116)inertia(G) = inertia(A) + inertia(C − B T A −1 B) (1117)When A=C =0, denoting by σ(B)∈ R m the nonincreasingly orderedsingular values of matrix B ∈ R m×m , then we have the eigenvalues[36,1.2, prob.17]([ 0 Bλ(G) = λB T 0])=[σ(B)−Ξσ(B)](1118)andinertia(G) = inertia(B T B) + inertia(−B T B) (1119)where Ξ is the order-reversing permutation matrix defined in (1313).