Some further matrix extensions of the Cauchy-Schwarz and Kantorovich inequalities, with some statistical applications Original Research Article

The well-known Cauchy-Schwarz and Kantorovich inequalities may be expressed in terms of vectors and a positive definite matrix. We consider what happens to these inequalities when the vectors are replaced by matrices, the positive definite matrix is allowed to be positive semidefinite singular, and the usual inequalities are replaced by Löwner partial orderings. Some examples in the context of linear statistical models are presented.