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

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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.

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Acknowledgements

The authors would like to thank the editor and two referees for their detailed comments. Discussions with Prof. C. Ling, Prof. Ph. Toint, Prof. Z. Huang along with his team members, Dr. W. Ding, Dr. Z. Luo, Dr. X. Wang, and Mr. C. Mo are very helpful.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Yun Miao

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Liqun Qi

  3. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Yimin Wei

Authors
  1. Yun Miao
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  2. Liqun Qi
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  3. Yimin Wei
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Corresponding author

Correspondence to Yimin Wei.

Additional information

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.

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Miao, Y., Qi, L. & Wei, Y. T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product. Commun. Appl. Math. Comput. 3, 201–220 (2021). https://doi.org/10.1007/s42967-019-00055-4

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  • DOI: https://doi.org/10.1007/s42967-019-00055-4

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