Convexity and concavity theorems on positive matrices and applications to geometric means and Khatri-rao products - Kilicman, Adem; Al Zhour, Zeyad Abdel Al Ziz (2005)
This paper is concerned with two generalizations of Ando’s geometric mean and two related generalizations of the Tracy-Singh product for partitioned matrices. We recover the relationship between the Ando’s geometric mean and the Kronecker product to the case of operator mean and Tracy-Singh product. We provide several operator inequalities associated with non-negative linear maps by means of concavity and convexity theorems. We apply the concavity and convexity theorems in order to obtain new unusual estimates for the Khatri-Rao product positive definite matrices. Finally, the results lead inequalities involving Hadamard product and Ando’s (α − power) geometric mean, as a special case.