Fourth-order tensor Riccati equations with the Einstein product

ABSTRACT

In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the $\mathcal{M}$-tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati equation from the matrix to the tensor case. The existence and uniqueness of the solution for the fourth-order $\mathcal{M}$-tensor Riccati equation are studied. Furthermore, we provide a perturbation analysis of the mixed and componentwise condition numbers for the $\mathcal{M}$-tensor Riccati equation. An algorithm for solving the equation is proposed. Numerical examples are presented to show the effectiveness of our algorithm.

KEYWORDS:

• Einstein product
• tensor Riccati equation
• $\mathcal{M}$-tensor
• tensor blocks
• spectrum
• condition number

AMS SUBJECT CLASSIFICATIONS:

• 15A48
• 15A69
• 65F10
• 65H10
• 65N22

Acknowledgments

The authors would like to thank the handling editor Ren-cang Li and Dr. Maolin Che for their useful discussion and suggestions on our paper and Professors Eric King-Wah Chu and Sanzheng Qiao for very detailed comments which greatly improve the presentation in our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Y. Miao is supported by the National Natural Science Foundation of China under grant 11771099. Y. Wei is supported by Innovation Program of Shanghai Municipal Education Commission. Z. Chen is supported by Key Project of Natural Science Foundation of Guizhou Province ([2020] 1Z002).