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Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

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Abstract

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (ℳ) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (ℳ) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.

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Acknowledgements

The research of Jun Ji was partly supported by the Kennesaw State University Tenured Faculty Professional Development Full Paid Leave Program in Fall 2015; Yimin Wei was supported by the International Cooperation Project of Shanghai Municipal Science and Technology Commission (Grant No. 16510711200).

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Authors and Affiliations

  1. Department of Mathematics, Kennesaw State University, Marietta, GA, 30060, USA

    Jun Ji

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

    Yimin Wei

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  1. Jun Ji
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  2. Yimin Wei
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Correspondence to Yimin Wei.

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Ji, J., Wei, Y. Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product. Front. Math. China 12, 1319–1337 (2017). https://doi.org/10.1007/s11464-017-0628-1

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  • DOI: https://doi.org/10.1007/s11464-017-0628-1

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