v2009.01.01 - Convex Optimization

574 APPENDIX A. LINEAR ALGEBRA
A.7.5
Zero definite
The domain over which an arbitrary real matrix A is zero definite can exceed
its 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 ) (1480)
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 ). For
any positive definite matrix A (for A +A T ≻ 0)
Further, [344,3.2, prob.5]
while
{x | x T Ax = 0} = 0 (1481)
{x | x T Ax = 0} = R M ⇔ A T = −A (1482)
{x | x H Ax = 0} = C M ⇔ A = 0 (1483)
The positive semidefinite matrix
[ ] 1 2
A =
0 1
for example, has no nullspace. Yet
(1484)
{x | x T Ax = 0} = {x | 1 T x = 0} ⊂ R 2 (1485)
which is the nullspace of the symmetrized matrix. Symmetric matrices are
not spared from the excess; videlicet,
[ ] 1 2
B =
(1486)
2 1
has eigenvalues {−1, 3} , no nullspace, but is zero definite on A.19
X ∆ = {x∈ R 2 | x 2 = (−2 ± √ 3)x 1 } (1487)
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 defined
lim
ε→0 +{x∈ R2 | x T Bx = ε} = X