Tensor distributions with covariance tensor or correlation tensor

TL;DR

This paper introduces tensor distributions, including covariance and correlation tensors, and explores their properties, defining tensor normal and elliptical distributions with proofs of their equivalences.

Contribution

It presents new definitions and properties of tensor distributions, extending classical concepts to tensor-valued data with formal proofs.

Findings

Defined the determinant of a tensor and its properties

Introduced the covariance and correlation tensors with their properties

Established the equivalence of tensor elliptical distribution representations

Abstract

In this article, we define the matricization of a tensor and we present some properties of the matricization. After that, we define the determinant of a tensor and we present some properties of the determinant. We define the covariance tensor and we present some properties of the covariance tensor. In a similar way, we define the correlation tensor. We define the tensor normal distribution. In a similar way, we define the tensor elliptical distributions. We prove the equivalence of the tensor elliptical distribution representations.