The equivalence of two partial orders on a convex cone of positive semidefinite matrices

Abstract

The Loewner partial order ≧ is defined on the space of Hermitian matrices by A ≧ B if A − B is positive semidefinite. Given a strictly increasing function f: ( a, b) → R, we define the partial order ≧ f on the set of Hermitian matrices with spectrum contained in ( a, b) by A ≧ fB if f(A) ≧ f(B). We say that the partial orders ≧ and ≧ f are equivalent ona set S of Hermitian matrices if A ≧ B if and only if A ≧ fB for all A, B ϵ S . It is clear that if the cone C is commutative, i.e., AB = BA for all A, B ϵ C , then the two partial orders are equivalent. Stepniak 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 we characterize the convex cones C of positive semidefinite matrices on which ≧ and ≧ f are equivalent for a class of functions that includes f( t) = t p , p > 1, and f( t) = e t . We introduce the class of strongly monotone matrix functions and prove the following result of independent interest: Let ƒ 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 semidefinite and that A and B have no common eigenvectors. Then f( A) − f( B) is positive definite. We also show that the functions f( t) = t p , 0 < p < 1, and f( t) = log t are strongly monotone of all orders. We also consider the partial order ≧ t 2. In this special case it is possible to obtain the results using more elementary techniques. We prove some results about the partial orders ≧ t p and ≧ exp. We conclude with two open questions.