H-tensors' Criteriaand Their Related Properties

The H-matrix is an important part of the matrix theory,which has a wide range of applications.Recently,the concept has been extended to higher order H-tensor.The H-tensor plays an important role in scientific computing and engineering applications,and it has important application value in the stability of automatic control systems,global optimization of polynomials,and noise reduction of medical images.In this paper,we will study the H-tensors' criteria and their relatd properties.First,we study the determinant conditions of M-tensors,and compare the special structures of M-tensors and H-tensors,then find the conditions for determining Htensors.Secondly,we study the non-negative H-tensors and other H-tensors with special structures,then explore their related properties.We proved that the H-tensor with special structure satisfies the linear property of the tensor addition and the number multiplication.Then We proved the irreducible H-tensor with positive diagonal element is a semi-nonnegative tensor,and its principal sub-tensors are nonsingular H-tensor.Finally,we study the basic decomposition of even an order symmetric tensor,analyze the basic part and the nonbasic part of the tensor.It is proved that a tensor is a semi-nonnegative tensor if the basic part is semi-nonnegative tensor.We also proved that a H-tensor,if it have basic decomposition,then its basic part of the tensor is also Htensor which get a necessary condition for the H-tensor.