Tensor Unfolding Characterization

Abstract

Tensors play a pivotal role in the realms of science and engineering, particularly in the realms of data analysis, machine learning, and computational mathematics. The process of unfolding a tensor into matrices, commonly known as tensor unfolding or matricization, serves as a valuable technique for simplifying the representation of tensors with higher orders. In this study, we initially derive unfolded matrices from a specified tensor over a B{'e}zout ring using a matrix equivalence relation. We proceed to elucidate the relationships between eigenvalues and eigenvectors within these unfolded matrices. Additionally, we employ the localization approach outlined by Gerstein to ascertain the count of distinct matrix equivalence classes present among the unfolded matrices.