A binomial expansion formula for weighted geometric means of unipotent matrices

Abstract

Following the Kubo-Ando theory of operator means we consider the weighted geometric mean $A{\mathrm{\#}}_{t}B$ of $n\times n$ upper triangular matrices A and B whose main diagonals are all 1, named the upper unipotent matrices. We also present its binomial expansion $A{\mathrm{\#}}_{t}B=\sum _{k=0}^{n-1}\left(\begin{array}{c}t\\ k\end{array}\right)A(A^{-1}B-I{)}^k,t\in \mathbb{R}.$ Showing that the weighted geometric mean is a geodesic of symmetry in the symmetric space equipped with point reflection, known as the Loos symmetric space, we derive several binomial identities on the Lie group of upper unipotent (resp. the Lie algebra of nilpotent) matrices.

Keywords:

• Unipotent matrix
• weighted geometric mean
• log-Euclidean mean
• binomial expansion
• loos symmetric space

2020 Mathematics Subject Classifications:

• 15B30
• 22E25
• 65F45

Acknowledgments

The authors are grateful to Professor Bruno Iannazzo for suggesting a formula for $A\mathrm{\#}B$ of $5\times 5$ upper unipotent matrices. All authors equally contributed to this work. No potential competing interest was reported by the authors.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of H. Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) [grant number 2020R1C1C1A01009185]. The work of S. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [grant number NRF-2022R1A2C4001306]. The work of Y. Lim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) [grant numbers 2015R1A3A2031159 and 2016R1A5A1008055].