On the Ando–Li–Mathias mean and the Karcher mean of positive definite matrices

Abstract

In this paper, from the viewpoint of the Ando–Hiai inequality, we make a comparison among three geometric means: The Ando–Li–Mathias geometric mean, the Karcher mean and the chaotic geometric mean of positive definite matrices. Among others, we show complements of the -variable Ando–Hiai inequality for the Ando–Li–Mathias geometric mean by means of the Kantorovich constant.

Keywords:

• Ando–Li–Mathias geometric mean
• Karcher mean
• chaotic geometric mean
• matrix geometric mean
• unitarily invariant norm
• Specht ratio
• Kantorovich constant
• Ando–Hiai inequality

MSC classification:

• 47A30
• 47A63
• 47A64

Acknowledgements

The authors would like to express their cordial thanks to the referee for his/her valuable suggestions and comments on Theorem 4.2.