Real tensor factorizations and generalized inverses under the $t$-product

TL;DR

This paper explores conditions under which tensor factorizations and generalized inverses in the $t$-product framework can be realized over the real numbers, extending complex-based methods.

Contribution

It characterizes structural conditions for real realizations of tensor factorizations and inverses, enabling transfer from complex to real tensor algebra.

Findings

Conjugate-pairing structure determines real realizability of tensors.

Provides real versions of several tensor factorizations.

Analyzes existence and structure of real generalized inverses.

Abstract

The algebraic theory of third-order tensors under the $t$-product is naturally formulated over the complex field via Fourier block diagonalization. However, many applications require real-valued representations. In this paper, we investigate structural conditions ensuring that tensor factorizations and generalized inverses admit real realizations. We show that these conditions can be characterized through the conjugate-pairing structure of the Fourier frontal slices, which determines when transform-domain constructions yield real tensors after inverse transformation. As applications, we obtain real versions of several tensor factorizations and analyze the existence and structure of associated generalized inverses. These results provide a framework for transferring matrix-based constructions to real tensors while preserving the algebraic constraints of the $t$-product.