Preservers of Partial Orders on the Set of all Variance-Covariance Matrices

Let Hn+ (R) be the cone of all positive semidefinite n x n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the Lowner and the minus partial orders. Motivated by applications in statistics we study these partial orders on Hn+ (R). We describe the form of all surjective maps on Hn+ (R), n > 1, that preserve the Lowner partial order in both directions. We present an equivalent definition of the minus partial order on Hn+ (R) and also characterize all surjective, additive maps on Hn+ (R), n >_ 3, that preserve the minus partial order in both directions.