The Equivalence of Two Partial Orders on A Convex Cone

The Loewner partial order is deened on the space of Hermitian matrices by A B if A ? B is positive semideenite. Given a strictly increasing function f :(a; b) ! R we deene the partial order f on the set of Hermitian matrices with spectrum contained in (a; b) by A f B if f(A) f(B): We say that the partial orders and f are equivalent on a set S of Hermitian matrices if A B if and only if A f B for all A; B 2 S: It is clear that if the cone C is commutative, i.e., AB = BA for all A; B 2 C, then the two partial orders are equivalent. Stepniak 7] conjectured the converse for the function f(t) = t 2 , and proved it for n 3. We provide a counterexample to Stepniak's conjecture for n 4 and in Theorem 4.4 we characterize the convex cones C of positive semideenite matrices on which and f are equivalent for a class of functions that includes f(t) = t p ; p > 1, and f(t) = exp t. We introduce the class of strongly monotone matrix functions in Section 3 and in Theorem 3.5 prove the following result of independent interest: Let f be a strongly monotone matrix function of order n and suppose that A and B are n-by-n Hermitian matrices such that A ? B is positive semideenite and that A and B have no common eigenvectors. Then f(A) ? f(B) is positive deenite. We also show that the functions f(t) = t p ; 0 < p < 1 and f(t) = log(t) are strongly monotone of all orders. In Section 5 we consider the partial order t 2. In this special case it is possible to obtain the results in Section 4 using more elementary techniques. In Section 6 we prove some results about the partial orders t p and exp. We conclude with two open questions.