Further results on Moore–Penrose inverses of tensors with application to tensor nearness problems

Reverse-order law for weighted Moore–Penrose inverse of tensors

“…The next result confirms that whenever the tensor A satisfies M * N A † MN = (M * N A) H , then (15) and (16) are sufficient to hold the reverse-order law.…”

“…Then the authors find the minimum-norm least-squares solution of some multilinear systems by using the same study and proposed different types of generalized inverses of tensors. In 2018, Panigrahy and Mishra [18] improved the definition of the Moore-Penrose inverse of an evenorder tensor to a tensor of any order via the same product which also appeared in [16] and [22]. Panigrahy and Mishra [18] also introduced an extension of the Moore-Penrose inverse of a tensor called as the Product Moore-Penrose inverse.…”

“…Definition 1.1. (Definition 3.3, [16] & Definition 1.1, [18]) Let X ∈ C I 1 ×···×I M ×J 1 ×···×J N . The tensor Y ∈ C J 1 ×···×J N ×I 1 ×···×I M satisfying the following four tensor equations:…”

Further results on the Drazin inverse of even‐order tensors

“…This article will concentrate on one such class of generalized inverse, namely, Drazin inverse, which was proposed Drazin 6 in 1958 in the context of rings and semigroups. The Drazin inverse plays an important role in various areas such as singular differential, 8 difference equations, 9 investigation of Cesaro‐Neumann iterations, 10 matrix splitting, 11 finite Markov chains, 9 and cryptography 12 . In particular, the spectral properties 13 of this inverse plays significant role in many applications 3 .…”

“…This interpretation is extended to the Moore‐Penrose inverse of tensors in References 20,21 and the solution of multilinear systems are discussed therein. Furthermore, its application to tensor nearness problems is demonstrated in Reference 8. Using such theory of Einstein product, Liang et al 22 investigated necessary and sufficient conditions for the invertibility of tensors.…”