The class of $B_1$-matrices is a subclass of $P$-matrices and introduced as a generalization of $B$-matrices. In this paper, we present several properties for $B_1$-matrices. Then, the infinity norm upper bound for the inverse of $B_1$-matrices is obtained. Furthermore, the error bound for the linear complementarity problem of $B_1$-matrices is presented. Finally, some numerical examples are given to illustrate our results.

In this paper, a new proof that $SD{D}_1$ matrices is a subclass of $H$-matrices is presented, and some properties of $SD{D}_1$ matrices are obtained. Based on the new proof, some upper bounds of the infinity norm of inverse of $SD{D}_1$ matrices and $SDD$ matrices are given. Moreover, we show that these new bounds of $SDD$ matrices are better than the well-known Varah bound for $SDD$ matrices in some cases. In addition, some numerical examples are given to illustrate the corresponding results.

$\mathcal{H}$-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.