Fourth-order tensor Riccati equations with the Einstein product. Issue 10 (3rd July 2022)

• Record Type: Journal Article
• Authors: Miao, Yun; Wei, Yimin; Chen, Zhen
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 10(2022)
• Page Start: 1831
• Page End: 1853
• Publication Date: 2022-07-03
• Subjects: Einstein product -- tensor Riccati equation -- M-tensor -- tensor blocks -- spectrum -- condition number; 15A48 -- 15A69 -- 65F10 -- 65H10 -- 65N22; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1777248
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the 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 M -tensor Riccati equation are studied. Furthermore, we provide a perturbation analysis of the mixed and componentwise condition numbers for the M -tensor Riccati equation. An algorithm for solving the equation is proposed. Numerical examples are presented to show the effectiveness of our algorithm.