A new characterization of symmetric $H^+$-tensors and $M$-tensors

Abstract

In this work, we present a new characterization of symmetric H + -tensors. It is known that a symmetric tensor is an H + -tensor if and only if it is a generalized diagonally dominant tensor with nonnegative diagonal elements. By exploring the diagonal dominance property, we derive new necessary and sufficient conditions for a symmetric tensor to be an H + -tensor. Based on these conditions, we propose a new method that allows to check if a tensor is a symmetric H + -tensor in polynomial time. In particular, this allows to efficiently compute the minimum H-eigenvalue of tensors in the related and important class of Mtensors. Furthermore, we show how this result can be used to approximately solve polynomial optimization problems.