v2007.09.13 - Convex Optimization

514 APPENDIX A. LINEAR ALGEBRAA.7.5Zero definiteThe domain over which an arbitrary real matrix A is zero definite can exceedits left and right nullspaces. For any positive semidefinite matrix A∈ R M×M(for A +A T ≽ 0){x | x T Ax = 0} = N(A +A T ) (1374)because ∃R A+A T =R T R , ‖Rx‖=0 ⇔ Rx=0, and N(A+A T )= N(R).Then given any particular vector x p , x T pAx p = 0 ⇔ x p ∈ N(A +A T ). Forany positive definite matrix A (for A +A T ≻ 0)Further, [298,3.2, prob.5]while{x | x T Ax = 0} = 0 (1375){x | x T Ax = 0} = R M ⇔ A T = −A (1376){x | x H Ax = 0} = C M ⇔ A = 0 (1377)The positive semidefinite matrix[ ] 1 2A =0 1for example, has no nullspace. Yet(1378){x | x T Ax = 0} = {x | 1 T x = 0} ⊂ R 2 (1379)which is the nullspace of the symmetrized matrix. Symmetric matrices arenot spared from the excess; videlicet,[ ] 1 2B =(1380)2 1has eigenvalues {−1, 3}, no nullspace, but is zero definite on A.19X ∆ = {x∈ R 2 | x 2 = (−2 ± √ 3)x 1 } (1381)A.19 These two lines represent the limit in the union of two generally distinct hyperbolae;id est, for matrix B and set X as definedlimε→0 +{x∈ R2 | x T Bx = ε} = X