In this work, we present a new characterization of symmetric \(H^+\) -tensors, also referred as generalized diagonally dominant tensors with nonnegative diagonals. Namely, by exploring their 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 identify if a tensor is a symmetric \(H^+\) -tensor in polynomial time, by solving a power cone optimization problem. Further, we show how this result can be used to efficiently compute the minimum H-eigenvalue of symmetric M-tensors and to provide tighter lower bounds for the minimum H-eigenvalue of the Fan product of two symmetric M-tensors. Throughout the article, numerical experiments are used to benchmark and illustrate the applications of our results.
