Eigenvalues and Jordan Forms of Dual Complex Matrices

doi:10.1007/s42967-023-00299-1

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1225-1241.doi: 10.1007/s42967-023-00299-1

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Eigenvalues and Jordan Forms of Dual Complex Matrices

Liqun Qi^1,2, Chunfeng Cui³

1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China;
2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China;
3. LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing, 100191, China

收稿日期:2023-06-18 修回日期:2023-07-11 接受日期:2023-07-15 出版日期:2023-09-28 发布日期:2023-09-28
通讯作者: Chunfeng Cui,E-mail:chunfengcui@buaa.edu.cn E-mail:chunfengcui@buaa.edu.cn
作者简介:Liqun Qi, E-mail:maqilq@polyu.edu.hk
基金资助:
This work was supported by the National Natural Science Foundation of China (Nos. 12126608, 12131004) and the Fundamental Research Funds for the Central Universities (Grant No. YWF-22-T-204).

Eigenvalues and Jordan Forms of Dual Complex Matrices

Liqun Qi^1,2, Chunfeng Cui³

1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China;
2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China;
3. LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing, 100191, China

Received:2023-06-18 Revised:2023-07-11 Accepted:2023-07-15 Online:2023-09-28 Published:2023-09-28
Supported by:
This work was supported by the National Natural Science Foundation of China (Nos. 12126608, 12131004) and the Fundamental Research Funds for the Central Universities (Grant No. YWF-22-T-204).

摘要/Abstract

摘要： Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition. This definition is used in the research related with brain science. Under this definition, eigenvalues of dual complex matrices are defined. However, there are cases of dual complex matrices which have no eigenvalues or have infinitely many eigenvalues. We show that an n×n dual complex matrix is diagonalizable if and only if it has exactly n eigenvalues with n appreciably linearly independent eigenvectors. Hermitian dual complex matrices are diagonalizable. We present the Jordan form of a dual complex matrix with a diagonalizable standard part, and the Jordan form of a dual complex matrix with a Jordan block standard part. Based on these, we give a description of the eigenvalues of a general square dual complex matrix.

关键词: Dual complex numbers, Matrices, Eigenvalues, Diagonalization, Jordan form

Abstract: Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition. This definition is used in the research related with brain science. Under this definition, eigenvalues of dual complex matrices are defined. However, there are cases of dual complex matrices which have no eigenvalues or have infinitely many eigenvalues. We show that an n×n dual complex matrix is diagonalizable if and only if it has exactly n eigenvalues with n appreciably linearly independent eigenvectors. Hermitian dual complex matrices are diagonalizable. We present the Jordan form of a dual complex matrix with a diagonalizable standard part, and the Jordan form of a dual complex matrix with a Jordan block standard part. Based on these, we give a description of the eigenvalues of a general square dual complex matrix.

Key words: Dual complex numbers, Matrices, Eigenvalues, Diagonalization, Jordan form

中图分类号:

Liqun Qi, Chunfeng Cui. Eigenvalues and Jordan Forms of Dual Complex Matrices[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1225-1241.

参考文献

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[18] Zhang, F., Wei, Y.: Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices. Commun. Algebra 29(6), 2363-2375 (2001)

Eigenvalues and Jordan Forms of Dual Complex Matrices

Eigenvalues and Jordan Forms of Dual Complex Matrices

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