Improvements on Geometric Means Related to the Tracy-Singh Products of Positive Matrices

Abstract

In this paper, a family of geometric means for positive matrices is studied; we discuss possible definitions of the geometric means of positive matrices, and some counter examples are given. It is still an open problem to find a completely satisfactory definition. Other problems are how to define the geometric, arithmetic, harmonic, $\alpha-$power and operator means of finitely many positive matrices. We generalize these means of two positive matrices to arrive the definitions of the weighted means of $k$ positive matrices. We recover and develop the relationship between the Ando's geometric mean and the Kronecker product to the Tracy-Singh product and other means. Some new attractive inequalities for the Tracy-Singh product, Khatri-Rao product and geometric means of several positive matrices are established. The results lead to the case of Kronecker and Hadamard products of any finite numbers of matrices. Keywords: Tracy-Singh product; Khatri-Rao product; Kronecker product; Hadamard product; Positive definite matrix; Geometric means; weighted geometric means.