@article{3196489,
    title = "One-sided weighted outer inverses of tensors",
    author = "Mosic, Dijana and Stanimirovic, Predrag S. and Sahoo, Jajati Keshari and and Behera, Ratikanta and Katsikis, Vasilios N.",
    journal = "JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS",
    year = "2021",
    volume = "388",
    publisher = "Elsevier",
    issn = "0377-0427",
    doi = "10.1016/j.cam.2020.113293",
    keywords = "Multidimensional array; Outer inverse; Tensor rank; Tensor equation;
Einstein product",
    abstract = "In this paper, for the first time in literature, we introduce one-sided
weighted inverses and extend the notions of one-sided inverses, outer
inverses and inverses along given elements. Although our results are new
and in the matrix case, we decided to present them in tensor space with
reshape operator. For this purpose, a left and right (M, N)-weighted (B,
C)-inverse and the (M, A)-weighted (B, C)-inverse of a tensor are
defined. Additionally, necessary and sufficient conditions for the
existence of these new inverses are presented. We describe the sets of
all left (or right) (M, A)-weighted (B, C)-inverses of a given tensor.
As consequences of these results, we consider the one-sided (B,
C)-inverse, (B, C)-inverse, one-sided inverse along a tensor and inverse
along a tensor. Further, we introduce a (M, A)-weighted (B, C)-outer
inverse and a W-weighted (B, C)-outer inverse of tensors with a few
characterizations. Then, corresponding algorithms for computing various
types of outer inverses of tensors are proposed, implemented and tested.
The prowess of the proposed inverses are demonstrated for finding the
solution of Poisson problem and the restoration of 3D color images. (C)
2020 Elsevier B.V. All rights reserved."
}