Computing Tensor Generalized Bilateral Inverses

We introduce tensor generalized bilateral inverses (TGBIs) under the Einstein tensor product as an extension of generalized bilateral inverses (GBIs) in the matrix environment. Moreover, the TBGI class includes so far considered composite generalized inverses (CGIs) for matrices and tensors. Applications of TBGIs for solving multilinear systems are presented. The characterizations and representations of the TGBI were studied and verified using a specific algebraic approach. Further, a few characterizations of known CGIs (such as the CMP, DMP, MPD, MPCEP, and CEPMP) are derived. The main properties of the TGBIs were exploited and verified through numerical examples.