On the extensions of the GD inverse of tensors via the M-Product

TL;DR

This paper extends the concept of the GD tensor inverse using the M-product, introducing new types of inverses, their properties, algorithms, and applications to multilinear equations, with numerical examples.

Contribution

It introduces the tensor GD inverse, GDMP inverse, and GD-Star inverse under the M-product, along with their properties, laws, algorithms, and applications to multilinear equations.

Findings

Established properties and representations of the GD inverse.

Developed algorithms for the GDMP inverse.

Applied the inverses to solve multilinear equations.

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.