T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

doi:10.1007/s42967-019-00055-4

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (2): 201-220.doi: 10.1007/s42967-019-00055-4

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T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

Yun Miao¹, Liqun Qi², Yimin Wei³

1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China;
3 School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China

收稿日期:2019-09-06 修回日期:2019-11-17 出版日期:2021-06-20 发布日期:2021-05-26
通讯作者: Yimin Wei, Yun Miao, Liqun Qi E-mail:ymwei@fudan.edu.cn,yimin.wei@gmail.com;15110180014@fudan.edu.cn;maqilq@polyu.edu.hk
基金资助:
Y. Miao is supported by the National Natural Science Foundation of China (Grant No. 11771099). L. Qi is supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717). Y. Wei is supported by the Innovation Program of Shanghai Municipal Education Commission.

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

Yun Miao¹, Liqun Qi², Yimin Wei³

1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China;
3 School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China

Received:2019-09-06 Revised:2019-11-17 Online:2021-06-20 Published:2021-05-26
Contact: Yimin Wei, Yun Miao, Liqun Qi E-mail:ymwei@fudan.edu.cn,yimin.wei@gmail.com;15110180014@fudan.edu.cn;maqilq@polyu.edu.hk
Supported by:
Y. Miao is supported by the National Natural Science Foundation of China (Grant No. 11771099). L. Qi is supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717). Y. Wei is supported by the Innovation Program of Shanghai Municipal Education Commission.

摘要/Abstract

摘要： In this paper, we investigate the tensor similarity and propose the T-Jordan canonical form and its properties. The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed. As a special case, we present properties when two tensors commute based on the tensor T-product. We prove that the Cayley-Hamilton theorem also holds for tensor cases. Then, we focus on the tensor decompositions: T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained. When an F-square tensor is not invertible with the T-product, we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases. The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form. The polynomial form of the T-Drazin inverse is also proposed. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.

关键词: T-Jordan canonical form, T-function, T-index, Tensor decomposition, T-Drazin inverse, T-group inverse, T-core-nilpotent decomposition

Abstract: In this paper, we investigate the tensor similarity and propose the T-Jordan canonical form and its properties. The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed. As a special case, we present properties when two tensors commute based on the tensor T-product. We prove that the Cayley-Hamilton theorem also holds for tensor cases. Then, we focus on the tensor decompositions: T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained. When an F-square tensor is not invertible with the T-product, we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases. The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form. The polynomial form of the T-Drazin inverse is also proposed. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.

Key words: T-Jordan canonical form, T-function, T-index, Tensor decomposition, T-Drazin inverse, T-group inverse, T-core-nilpotent decomposition

中图分类号:

Yun Miao, Liqun Qi, Yimin Wei. T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product[J]. Communications on Applied Mathematics and Computation, 2021, 3(2): 201-220.

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T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

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