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The Moore–Penrose inverse of tensors via the M-product

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Abstract

This paper studies the issues about the Moore–Penrose inverse of tensors with the M-product. The goal of this paper is threefold. Firstly, we define the Moore–Penrose inverse of tensors under the M-product, and obtain several formulas for the Moore–Penrose inverse of tensors. In addition, we study the least square and minimum-norm solutions of the tensor equation. The conditions for the solutions to be the least square or minimum-norm solutions are given. Finally, we discuss the necessary and sufficient conditions for the reverse order law of the generalized inverses of tensors based on the M-product.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments, which have significantly improved the paper.

Funding

This work was supported by Talent Introduction and Scientific Research Start-Up Project of Guangxi Minzu University (No.2021KJQD02), the Special Fund for Science and Technological Bases and Talents of Guangxi (No. GUIKE21220024), the National Natural Science Foundation of China (No.12061015), the National Natural Science Foundation of China (No. 11571098) and the Guangxi Natural Science Foundation (No.2018GXNS FDA281023).

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Authors and Affiliations

  1. School of Mathematics and physics, Guangxi Minzu University, Nanning, 530006, People’s Republic of China

    Hongwei Jin, Shumin Xu & Yuzhen Wang

  2. School of Education, Guangxi Vocational Normal University, Nanning, 530006, People’s Republic of China

    Xiaoji Liu

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  1. Hongwei Jin
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  2. Shumin Xu
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  3. Yuzhen Wang
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  4. Xiaoji Liu
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Correspondence to Hongwei Jin.

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Communicated by yimin wei.

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Jin, H., Xu, S., Wang, Y. et al. The Moore–Penrose inverse of tensors via the M-product. Comp. Appl. Math. 42, 294 (2023). https://doi.org/10.1007/s40314-023-02427-2

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