Positive Semi-Definite matrices and subtraction I have been wondering about this for some time, and I haven't been able to answer the question myself. I also haven't been able to find anything about it on the internet. So I will ask the question here:
Question: Assume that $A$ and $B$ both are positive semi-definite. When is $C = (A-B)$ positive semi-definite?
I know that I can figure it out for given matrices, but I am looking for a necessary and sufficient condition.
It is of importance when trying to find solutions to conic-inequality systems, where the cone is the cone generated by all positive semi-definite matrices. The question I'm actually interested in finding nice result for are:
Let $x \in \mathbb{R}^n$, and let $A_1,\ldots,A_n,B$ be positive semi-definite. When is
$(\sum^n_{i=1}x_iA_i) - B$
positive semi-definite? 
I feel the answer to my first question should yield the answer to the latter. I am looking for something simpler than actually calculating the eigenvalues.

There's no easy answer. The relation is usually denoted $B\leq A$. The most useful test I can think of is that $A\leq B$ if and only if
$$
x^TBx\leq x^TAx, \ \ x\in\mathbb{R}^n.
$$
