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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 ageneralized diagonally dominant tensor with nonnegative diagonal elements. Byexploring the diagonal dominance property, we derive new necessary andsufficient conditions for a symmetric tensor to be an $H^+$-tensor. Based onthese conditions, we propose a novel method that allows to check if a tensor isa symmetric $H^+$-tensor in polynomial time. Moreover, these results can beapplied to the closely related and important class of $M$-tensors. Inparticular, this allows to efficiently compute the minimum $H$-eigenvalue ofsymmetric $M$-tensors. Furthermore, we show how this latter result can be usedto provide tighter lower bounds for the minimum $H$-eigenvalue of the Fanproduct of two symmetric $M$-tensors.

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