Tensor inversion and its application to the tensor equations with Einstein product

ABSTRACT

Recently, the inverse of an even-order square tensor has been put forward in [Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl. 2013;34(2):542–570] by means of the tensor group consisting of even-order square tensors equipped with the Einstein product. In this paper, several necessary and sufficient conditions for the invertibility of a tensor are obtained, and some approaches for calculating the inverse (if it exists) are proposed. Furthermore, the Cramer's rule and the elimination method for solving the tensor equations with the Einstein product are derived. In addition, the tensor eigenvalue problem mentioned in [Qi L-Q. Theory of tensors (hypermatrices). Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University; 2014] can also be addressed by using the elimination method mentioned above. By the way, the LU decomposition and the Schur decomposition of matrices are extended to tensor case. Numerical examples are provided to illustrate the main results.

COMMUNICATED BY:

• Nung-Sing Sze

KEYWORDS:

• Elementary tensor transformations
• tensor rank
• tensor determinant
• tensor inversion
• tensor equations

AMS SUBJECT CLASSIFICATION (2000):

• 15A69
• 65F05
• 65F15
• 65F40

Acknowledgments

The authors are grateful to the referees and the editor, whose valuable suggestions led to major improvements over the draft of this paper. The first author is indebted to Dr Zhen Chen and Dr Wei-yang Ding for their warm help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11571004]. The research of the first author was also supported by the Science Foundation of Education Department of Gansu Province [grant number 2017A-078], and Tianshui Normal University [grant number TAS1603] as well as the Key Discipline Construction Foundation of Tianshui Normal University. The research of the third author was also supported by the Fundamental Research Funds for the Central Universities [grant number lzujbky-2017-it54].