%0 Journal Article
%T Improvements on Geometric Means Related to the Tracy-Singh Products of Positive Matrices
%A Adem Kilicman
%A Zeyad Al Zhour
%J Matematika
%D 2006
%I Universiti Teknologi Malaysia
%X 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, ¦Á 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.
%K Tracy-Singh product
%K Khatri-Rao product
%K Kronecker product
%K Hadamard product
%K Positive definite matrix
%K Geometric means
%K weighted geometric means.
%U http://www.fs.utm.my/matematika/images/stories/matematika/20062216.pdf