Question

Sufficient condition for difference of two positive semidefinite matrices to be positive semidefinite

Answer 1

Answer:

am relatively new to linear algebra, and have been struggling with a problem for a few days now. Say we have two positive semi-definite matrices A and B, and further assume that A and B are such that A−B is also positive semi-definite. Can it be shown that det(A)≥det(B)? In my own attempts, I can see that Tr(A)≥Tr(B), but I do not think this is enough to prove the desired result. Perhaps there is something to be said about the relative magnitudes of the eigenvalues of A and B, but I can't see it. In any case, I would appreciate any help. Thanks a lot.