A New Class of Positive Semi-definite Tensors

TL;DR

This paper introduces MO tensors, a new class of positive semi-definite tensors inspired by Moler matrices, and explores their special cases, proving positive definiteness and analyzing eigenvalues.

Contribution

It presents a novel class of positive semi-definite tensors, including Sup-MO and essential MO tensors, with new theoretical properties and eigenvalue behaviors.

Findings

Sup-MO tensor has a positive smallest H-eigenvalue that approaches zero as dimension increases.

Essential MO tensor is proven to be a completely positive tensor.

The paper establishes positive definiteness of special MO tensor cases.

Abstract

In this paper, a new class of positive semi-definite tensors, the MO tensor, is introduced. It is inspired by the structure of Moler matrix, a class of test matrices. Then we focus on two special cases in the MO-tensors: Sup-MO tensor and essential MO tensor. They are proved to be positive definite tensors. Especially, the smallest H-eigenvalue of a Sup-MO tensor is positive and tends to zero as the dimension tends to infinity, and an essential MO tensor is also a completely positive tensor.