Extensions of Monotonicity Results to Semisimple Lie Groups

Abstract

We extend a monotonicity result of Wang and Gong on the product of powers of positive definite matrices. This result concerns the eigenvalues of such products which written as vectors have a log majorization relationship. Our extension is in the context of semisimple Lie groups which relies on the complete multiplicative Jordan decomposition and which we express with Kostant’s preorder. An analogous result on singular values is also obtained.