数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (3): 659-669.doi: 10.1007/s10473-020-0305-4

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LIE-TROTTER FORMULA FOR THE HADAMARD PRODUCT

王静1, 李永刚2, 孙华飞3   

  1. 1 School of Information, Beijing Wuzi University, Beijing 101149, China;
    2 College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, China;
    3 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
    4 Beijing Key Laboratory on MCAACI, Beijing 100081, China
  • 收稿日期:2018-10-24 修回日期:2019-05-17 出版日期:2020-06-25 发布日期:2020-07-17
  • 通讯作者: Jing WANG E-mail:wangjingzzumath@163.com
  • 作者简介:Yonggang LI,E-mail:liyonggang914@126.com;Huafei SUN,E-mail:huafeisun@bit.edu.cn
  • 基金资助:
    H. Sun is supported by NSFC (61179031); J. Wang is supported by General Project of Science and Technology Plan of Beijing Municipal Education Commission (KM202010037003).

LIE-TROTTER FORMULA FOR THE HADAMARD PRODUCT

Jing WANG1, Yonggang LI2, Huafei SUN3   

  1. 1 School of Information, Beijing Wuzi University, Beijing 101149, China;
    2 College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, China;
    3 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
    4 Beijing Key Laboratory on MCAACI, Beijing 100081, China
  • Received:2018-10-24 Revised:2019-05-17 Online:2020-06-25 Published:2020-07-17
  • Contact: Jing WANG E-mail:wangjingzzumath@163.com
  • Supported by:
    H. Sun is supported by NSFC (61179031); J. Wang is supported by General Project of Science and Technology Plan of Beijing Municipal Education Commission (KM202010037003).

摘要: Suppose that A and B are two positive-definite matrices, then, the limit of (Ap/2BpAp/2)1/p as p tends to 0 can be obtained by the well known Lie-Trotter formula. In this article, we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative, and obtain the explicit formula of the limit (Ap * Bp)1/p as p tends to 0. Furthermore, the existence of the limit of (Ap * Bp)1/p as p tends to +∞ is proved.

关键词: Lie-Trotter formula, reciprocal Lie-Trotter formula, Hadamard product, positive-definite matrix

Abstract: Suppose that A and B are two positive-definite matrices, then, the limit of (Ap/2BpAp/2)1/p as p tends to 0 can be obtained by the well known Lie-Trotter formula. In this article, we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative, and obtain the explicit formula of the limit (Ap * Bp)1/p as p tends to 0. Furthermore, the existence of the limit of (Ap * Bp)1/p as p tends to +∞ is proved.

Key words: Lie-Trotter formula, reciprocal Lie-Trotter formula, Hadamard product, positive-definite matrix

中图分类号: 

  • 15A42