Norms of Dual Complex Vectors and Dual Complex Matrices

doi:10.1007/s42967-022-00215-z

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (4): 1484-1508.doi: 10.1007/s42967-022-00215-z

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Norms of Dual Complex Vectors and Dual Complex Matrices

Xin-He Miao, Zheng-Hai Huang

School of Mathematics, Tianjin University, Tianjin 300354, China

收稿日期:2022-04-15 修回日期:2022-08-15 发布日期:2023-12-16
通讯作者: Zheng-Hai Huang,E-mail:huangzhenghai@tju.edu.cn;Xin-He Miao,E-mail:xinhemiao@tju.edu.cn E-mail:huangzhenghai@tju.edu.cn;xinhemiao@tju.edu.cn
基金资助:
The second author’s work is supported by the National Natural Science Foundation of China (Grant No. 11871051).

Norms of Dual Complex Vectors and Dual Complex Matrices

Xin-He Miao, Zheng-Hai Huang

School of Mathematics, Tianjin University, Tianjin 300354, China

Received:2022-04-15 Revised:2022-08-15 Published:2023-12-16
Contact: Zheng-Hai Huang,E-mail:huangzhenghai@tju.edu.cn;Xin-He Miao,E-mail:xinhemiao@tju.edu.cn E-mail:huangzhenghai@tju.edu.cn;xinhemiao@tju.edu.cn
Supported by:
The second author’s work is supported by the National Natural Science Foundation of China (Grant No. 11871051).

摘要/Abstract

摘要： In this paper, we investigate some properties of dual complex numbers, dual complex vectors, and dual complex matrices. First, based on the magnitude of the dual complex number, we study the Young inequality, the Hölder inequality, and the Minkowski inequality in the setting of dual complex numbers. Second, we define the p-norm of a dual complex vector, which is a nonnegative dual number, and show some related properties. Third, we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices. In particular, we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors, and give expressions of three important operator norms of dual complex matrices.

关键词: Dual complex number, Dual complex vector, Dual complex matrix, p-norm, Operator norm

Abstract: In this paper, we investigate some properties of dual complex numbers, dual complex vectors, and dual complex matrices. First, based on the magnitude of the dual complex number, we study the Young inequality, the Hölder inequality, and the Minkowski inequality in the setting of dual complex numbers. Second, we define the p-norm of a dual complex vector, which is a nonnegative dual number, and show some related properties. Third, we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices. In particular, we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors, and give expressions of three important operator norms of dual complex matrices.

Key words: Dual complex number, Dual complex vector, Dual complex matrix, p-norm, Operator norm

中图分类号:

Xin-He Miao, Zheng-Hai Huang. Norms of Dual Complex Vectors and Dual Complex Matrices[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1484-1508.

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Norms of Dual Complex Vectors and Dual Complex Matrices

Norms of Dual Complex Vectors and Dual Complex Matrices

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