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Perturbation theory for Moore–Penrose inverse of tensor via Einstein product

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Abstract

The spectral norm of an even-order tensor is defined and investigated. An equivalence between the spectral norm of tensors and matrices is given. Using derived representations of some tensor expressions involving the Moore–Penrose inverse, we investigate the perturbation theory for the Moore–Penrose inverse of tensor via Einstein product. The classical results derived by Stewart (SIAM Rev 19:634–662, 1977) and Wedin (BIT 13:217–232, 1973) for the matrix case are extended to even-order tensors. An implementation in the Matlab programming language is developed and used in deriving appropriate numerical examples.

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Acknowledgements

The authors would like to thank the editor and two referees for their detailed comments. This research is supported by the bilateral project between China and Serbia “The theory of tensors, operator matrices and applications (No. 4-5)”. H. Ma would like to thank Prof. Dragana S. Cvetković Ilić for her kind invitation and great hospitality; thank Prof. D.S. Djordjević and Prof. V. Rakoćević for their nice monograph (Djordjević and Rakočević 2008). Partial work is completed during her visiting at University of Nis̆ in 2017.

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Authors and Affiliations

  1. School of Mathematical Science, Harbin Normal University, Harbin, 150025, People’s Republic of China

    Haifeng Ma & Na Li

  2. Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, Niš, 18000, Serbia

    Predrag S. Stanimirović

  3. National and Kapodistrian University of Athens, Sofokleous 1 Street, 10559, Athens, Greece

    Vasilios N. Katsikis

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  1. Haifeng Ma
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  2. Na Li
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  3. Predrag S. Stanimirović
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  4. Vasilios N. Katsikis
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Correspondence to Predrag S. Stanimirović.

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Communicated by Jinyun Yuan.

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Haifeng Ma is supported by the bilateral project between China and Poland (No. 37-18). Predrag S. Stanimirović gratefully acknowledges support from the Research Project 174013 of the Serbian Ministry of Science.

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Ma, H., Li, N., Stanimirović, P.S. et al. Perturbation theory for Moore–Penrose inverse of tensor via Einstein product. Comp. Appl. Math. 38, 111 (2019). https://doi.org/10.1007/s40314-019-0893-6

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  • DOI: https://doi.org/10.1007/s40314-019-0893-6

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