On semi-definiteness and minimal H-eigenvalue of a symmetric space tensor using nonnegative polynomial optimization techniques

Verifying the positive semi-definiteness of a symmetric space tensor is an important and challenging topic in tensor computation. In this paper, we develop two methods to address the problem based on the theory of nonnegative polynomials which enables us to establish semi-definite programs to examine the positive semi-definiteness of a given symmetric space tensor. Moreover, using the similar idea, we can show that the minimal H-eigenvalue of a symmetric space tensor must be the optimal value of a semi-definite program. Computational results and discussions are provided to illustrate the significance of the results and the effectiveness of the proposed methods.