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Frontiers of Mathematics in China

, Volume 13, Issue 4, pp 893–911

# Generalized inverses of tensors via a general product of tensors

• Lizhu Sun
• Baodong Zheng
• Yimin Wei
• Changjiang Bu
Research Article

## Abstract

We define the {i}-inverse (i = 1, 2, 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.

## Keywords

Tensor generalized inverse tensor equation general product of tensor

## MSC

15A09 15A69 65F20 65F15

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## Notes

### Acknowledgements

The authors are very grateful to the referees for their valuable suggestions, which have considerably improved the paper. Yimin Wei was supported by the International Cooperation Project of Shanghai Municipal Science and Technology Commission (Grant No. 16510711200) and the National Natural Science Foundation of China (Grant No. 11771099); Changjiang Bu was supported by the National Natural Science Foundation of China (Grant No. 11371109).

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## Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Lizhu Sun
• 1
• Baodong Zheng
• 2
• Yimin Wei
• 3
• Changjiang Bu
• 1
1. 1.College of ScienceHarbin Engineering UniversityHarbinChina
2. 2.School of ScienceHarbin Institute of TechnologyHarbinChina
3. 3.School of Mathematical Sciences, Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiChina