Often, when there are two matrices $A$ and $B$ such that $A-B$ is positive definite, this relationship is written $A \gt B$, $A \succ B$ or similar. It has the advantage of being consistent with the usual $\gt$ relation when $A$ and $B$ are scalars.
The use of this notation is not completely transparent, so one ends up writing something like:
... where $A \gt B$ denotes the positive-definiteness of the matrix $A-B$.
thereby defeating the point of writing it in this concise way. One might as well just write "$A-B$ is positive definite" from the start. So, I was wondering if there was a more succinct way of describing the $\gt$ relationship in this case? I like the notation and would like to keep it, but I can't see how it can be justifiable unless one makes heavy use of it (which I don't intend to).
What about: $x^TAx\gt x^TBx$? This implies positive definiteness without the need to define it