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

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 novel method that allows to check if a tensor is a symmetric $H^+$-tensor in polynomial time. Moreover, these results can be applied to the closely related and important class of $M$-tensors. In particular, this allows to efficiently compute the minimum $H$-eigenvalue of symmetric $M$-tensors. Furthermore, we show how this latter result can be used to provide tighter lower bounds for the minimum $H$-eigenvalue of the Fan product of two symmetric $M$-tensors.