On the extensions of the GD inverse of tensors via the M-Product
Hongwei Jin, Siran Chen, Shaowu Huang, Predrag S. Stanimirović
TL;DR
This work extends tensor generalized inverses to the M-product setting by developing the GD inverse, its core-nilpotent decomposition, and a constructive algorithm, and by introducing the GDMP and GD-Star inverses with corresponding algorithms and order laws. It establishes reverse- and forward-order laws for these inverses and demonstrates how they solve multilinear systems, providing explicit solution forms and numerical illustrations. The contributions deliver a cohesive framework for tensor inversion under the M-product, enabling efficient handling of high-dimensional multilinear equations. The results have practical implications for multidimensional data processing where tensor algebra under the M-product is advantageous.
Abstract
We study extensions of the GD tensor inverse using the M-product. The aim of current research is threefold. In the first place, the tensor GD inverse under the M-product is introduced and considered. We give the several properties and representations of the GD inverse using the core nilpotent decomposition and then establish the reverse-order law rules for the GD inverse. Second, the tensor GDMP inverse is studied and the corresponding numerical algorithm is given. In addition, the reverse- and forward-order laws of the GDMP inverse are established. Third, the GD-Star tensor inverse under the M-product is introduced and studied. Finally, the GD inverse, GDMP inverse and GD-Star inverse solutions of multilinear equations are investigated. Illustrative numerical calculation is performed.