v2010.10.26 - Convex Optimization

614 APPENDIX A. LINEAR ALGEBRAA.4.0.1.1 Example. Sparse Schur conditions.Setting matrix A to the identity simplifies the Schur conditions. Oneconsequence relates definiteness of three quantities:[ ][ ]I BI 0B T ≽ 0 ⇔ C − B T B ≽ 0 ⇔C0 T C −B T ≽ 0 (1526)BA.4.0.1.2 Exercise. Eigenvalues λ of sparse Schur-form.Prove: given C −B T B = 0, for B ∈ R m×n and C ∈ S n⎧([ ]) ⎪⎨ 1 + λ(C) i , 1 ≤ i ≤ nI BλB T = 1, n < i ≤ mCi⎪⎩0, otherwise(1527)A.4.0.1.3 Theorem. Rank of partitioned matrices. [393,2.2 prob.7]When symmetric matrix A is invertible and C is symmetric,[ ] [ ]A B A 0rankB T = rankC 0 T C −B T A −1 B(1528)= rankA + rank(C −B T A −1 B)equals rank of main diagonal block A plus rank of its Schur complement.Similarly, when symmetric matrix C is invertible and A is symmetric,[ ] [ ]A B A − BCrankB T = rank−1 B T 0C0 T C(1529)= rank(A − BC −1 B T ) + rankCProof. The first assertion (1528) holds if and only if [202,0.4.6c][ ] [ ]A B A 0∃ nonsingular X,Y XB T Y =C 0 T C −B T A −1 (1530)BLet [202,7.7.6]Y = X T =[ I −A −1 B0 T I]⋄(1531)