Some results on the generalized inverse of tensors and idempotent tensors

TL;DR

This paper explores generalized inverses and idempotent tensors in the context of tensor algebra, providing new solutions to tensor equations, analyzing eigenvalues, and establishing relations between these concepts.

Contribution

It introduces new types of tensor inverses and idempotent tensors, and studies their properties and applications in solving tensor equations.

Findings

Solutions to tensor equations using generalized inverses

Eigenvalues of k-T-idempotent and idempotent tensors characterized

Relations between tensor inverses and idempotent tensors established

Abstract

Let $\mathcal{A}$ be an order $t$ dimension $m\times n\times \cdots \times n$ tensor over complex field. In this paper, we study some {generalized inverses} of $\mathcal{A}$, the {$k$-T-idempotent tensors} and the idempotent tensors based on the general tensor product. Using the tensor generalized inverse, some solutions of the equation $\mathcal{A}\cdot x^{t-1}=b$ are given, where $x$ and $b$ are dimension $n$ and $m$ vectors, respectively. The {generalized inverses} of some block tensors, the eigenvalues of {$k$-T-idempotent tensors} and idempotent tensors are given. And the relation between the generalized inverses of tensors and the $k$-T-idempotent tensors is also showed.