Within the field of multilinear algebra, inverses and generalized inverses of tensors based on the Einstein product have been investigated over the past few years. The notion of the weighted Moore–Penrose inverses of even-order tensors in the framework of the Einstein product was introduced recently (Ji and Wei in Front Math China 12(6):1319–1337, 2017). In this article, we introduce the weighted Moore–Penrose inverse of an arbitrary-order tensor. We also investigate the singular value decomposition and full-rank decomposition of arbitrary-order tensors using reshape operation. Derived representations are used for two purposes: (1) to obtain a few new characterizations and representations of weighted Moore–Penrose inverse of arbitrary-order tensors; (2) to explore various necessary and sufficient conditions for the reverse-order law for the inverse to hold. In addition to these, we discuss applications of singular value decomposition and the Moore–Penrose inverse of an arbitrary-order tensor to a few 3D color image processing.

在多线性代数领域，基于爱因斯坦积的张量的逆和广义逆在过去的几年中已得到研究。最近引入了爱因斯坦乘积框架中偶数张量的加权Moore-Penrose逆的概念（Ji和Wei在Front Math China 12（6）：1319-1337，2017）。在本文中，我们介绍了任意阶张量的加权Moore-Penrose逆。我们还研究了使用整形的任意阶张量的奇异值分解和满秩分解操作。派生的表示有两个目的：（1）获得一些新的特征和任意阶张量的加权Moore-Penrose逆的表示；（2）探索适用于逆定律的逆序定律的各种充要条件。除了这些，我们讨论了奇异值分解的应用以及任意阶张量的Moore-Penrose逆在少数3D彩色图像处理中的应用。