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

Abstract 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. Highlights • We find two methods to verify the positive semi-definiteness of a symmetric space tensor. • The first method could verify the positive semi-definiteness of a symmetric space tensor in polynomial time. • The second method could verify the positive semi-definiteness of a symmetric space tensor in the large order case. • We discuss the method how to solve the H-eigenvalue of a symmetric space tensor. [ABSTRACT FROM AUTHOR]