A new S-type eigenvalue localization set for tensors is given by partitioning the index set  into a subset Sand its complement. It is shown that the new set is tighter than those in Li  et al. [35]. Based on this new set, we give a checkable sufficient condition for the positive (semi-)definiteness of tensors and an upper bound for the spectral radius of nonnegative tensors.

tensor eigenvalue, localization set, positive definite, spectral radius.