Further results on $B$-tensors with application to the location of real eigenvalues

Abstract: In this paper, we give a further study on $B$-tensors and introduce doubly $B$-tensors that contain $B$-tensors. We show that they have similar properties, including their decompositions and strong relationship with strictly (doubly) diagonally dominated tensors. As an application, the properties of $B$-tensors are used to localize real eigenvalues of some tensors, which would be very useful in verifying the positive semi-definiteness of a tensor.