Extension of Moore–Penrose inverse of tensor via Einstein product

Abstract

In this paper, we first give an expression for the Moore-Penrose inverse of the product of two tensors via the Einstein product. We then introduce a new generalized inverse of a tensor called product Moore-Penrose inverse. A necessary and sufficient condition for the coincidence of the Moore-Penrose inverse and the product Moore-Penrose inverse is also proposed. Finally, the triple reverse order law of tensors is introduced.

Core and core-EP inverses of tensors

“…Representations for the weighted Moore-Penrose inverse of tensors were considered in [13]. Panigrahy and Mishra in [21] investigated the Moore-Penrose inverse of the Einstein product of two tensors. Ji and Wei [14] investigated the Drazin inverse of even-order square tensors under the Einstein product.…”

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

“…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.…”

“…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. The definition of the Moore-Penrose inverse of an arbitrary order tensor is recalled below.…”

“…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:…”

“…They also provided a method of computation of the weighted Moore-Penrose inverse using full rank decomposition of a tensor which was introduced by Liang and Zheng [16]. Among other results, they attempted the problem of triple reverse-order law mentioned in the conclusion section of [18] for the weighted Moore-Penrose inverse. In this paper, our aim is to study the reverse-order law for the weighted Moore-Penrose inverse of tensors via the Einstein product.…”