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Sum-of-squares (SOS) tensors plays an important role in tensor positive definiteness and polynomial optimization. So it is important to figure out what kind of tensors are SOS tensors. In this paper, we first show that several types of even order symmetric tensors are SOS tensors. The inclusive relation between several types of existing SOS tensors are developed under suitable conditions. By exploring the structure of LLK tensor, which is a special type of SOS tensor, we propose a numerical algorithm to check whether a given even order symmetric tensor is LLK tensor or not. Numerical experiments are also reported to show the efficiency of the proposed method.

Further results on sum-of-squares tensors