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Weighted Moore–Penrose inverses of arbitrary-order tensors

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Abstract

Within the field of multilinear algebra, inverses and generalized inverses of tensors based on the Einstein product have been investigated over the past few years. The notion of the weighted Moore–Penrose inverses of even-order tensors in the framework of the Einstein product was introduced recently (Ji and Wei in Front Math China 12(6):1319–1337, 2017). In this article, we introduce the weighted Moore–Penrose inverse of an arbitrary-order tensor. We also investigate the singular value decomposition and full-rank decomposition of arbitrary-order tensors using reshape operation. Derived representations are used for two purposes: (1) to obtain a few new characterizations and representations of weighted Moore–Penrose inverse of arbitrary-order tensors; (2) to explore various necessary and sufficient conditions for the reverse-order law for the inverse to hold. In addition to these, we discuss applications of singular value decomposition and the Moore–Penrose inverse of an arbitrary-order tensor to a few 3D color image processing.

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Acknowledgements

The authors would like to thanks the handling editor and referees for their detailed comments and suggestions. Ratikanta Behera is grateful for the supported by Science and Engineering Research Board (SERB), Department of Science and Technology, India, under the Grant No. EEQ/2017/000747. Ram N. Mohapatra is grateful to the Mohapatra Family Foundation and the College of Graduate Studies, University of Central Florida, Orlando, for their financial support for this research.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Ratikanta Behera & R. N. Mohapatra

  2. Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia, West Bengal, India

    Ratikanta Behera & Sandip Maji

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  1. Ratikanta Behera
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  2. Sandip Maji
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  3. R. N. Mohapatra
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Correspondence to Ratikanta Behera.

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Communicated by Ke Chen.

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Behera, R., Maji, S. & Mohapatra, R.N. Weighted Moore–Penrose inverses of arbitrary-order tensors. Comp. Appl. Math. 39, 284 (2020). https://doi.org/10.1007/s40314-020-01328-y

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