A new class of positive semi-definite tensors

Yi Xu,Liqun Qi,Jinjie Liu

doi:10.3934/jimo.2018186

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.

Highlights

In recent decades, tensors, as the natural extension of matrices, have been more and more ubiquitous in a wide variety of applications, such as data analysis and mining, signal processing, computational biology and so on [3, 6]
Completely positive tensors, which is connected with nonnegative tensor factorization, have significant applications in polynomial optimization problems, statistics, data analysis and so on
Inspired by the good properties of Moler matrix, we construct a new class of positive semi-definite tensors

Summary

Introduction

Tensors, as the natural extension of matrices, have been more and more ubiquitous in a wide variety of applications, such as data analysis and mining, signal processing, computational biology and so on [3, 6]. Completely positive tensors, which is connected with nonnegative tensor factorization, have significant applications in polynomial optimization problems, statistics, data analysis and so on. They were first introduced in [15]. In [9], two well-known classes of test matrices, Pascal matrices and Lehmer matrices were extended to Pascal tensors and Lehmer tensors They are checkable and were proved to be completely positive tensors [9]. There is another class of test matrices, the Moler matrices. Inspired by the good properties of Moler matrix, we construct a new class of positive semi-definite tensors.

Preliminaries

Main Results

Final Remarks