Two orderings on a convex cone of nonnegative definite matrices (Q580477)

Let \({\mathcal M}\) be a convex cone of real nonnegative definite matrices. For \(M_ 1,M_ 2\in {\mathcal M}\) denote \(M_ 1\geq M_ 2\) or \(M_ 1\succcurlyeq M_ 2\) if, respectively, \(M_ 1-M_ 2\) or \(M^ 2_ 1- M^ 2_ 2\) is nonnegative definite. Conditions for equivalence of the two orderings are given. In particular, if \(M_ 1M_ 2=M_ 2M_ 1\) for all \(M_ 1,M_ 2\in {\mathcal M}\), then the two orderings are equivalent on \({\mathcal M}\). The reverse direction of this result is posed as a conjecture.