Some new criteria for judging $\mathcal{H}$-tensors and their applications

$\mathcal{H}$-tensors play a key role in identifying the positive definiteness of even-order real symmetric tensors. Some criteria have been given since it is difficult to judge whether a given tensor is an $\mathcal{H}$-tensor, and their range of judgment has been limited. In this paper, some new criteria, from an increasing constant $k$ to scale the elements of a given tensor can expand the range of judgment, are obtained. Moreover, as an application of those new criteria, some sufficient conditions for judging positive definiteness of even-order real symmetric tensors are proposed. In addition, some numerical examples are presented to illustrate those new results.