The class of generalized $SD{D}_1\left(GSD{D}_1\right)$ matrices is a new subclass of $H$-matrices. In this paper, we focus on the subdirect sum of $GSD{D}_1$ matrices, and some sufficient conditions to ensure that the subdirect sum of $GSD{D}_1$ matrices with strictly diagonally dominant $\left(SDD\right)$ matrices is in the class of $GSD{D}_1$ matrices are given. Moreover, corresponding examples are given to illustrate our results.

Strong $\mathcal{H}$-tensors have many important applications in practical problems. In particular, strong $\mathcal{H}$-tensors play an important role in the positive qualitative determination of multivariate even-order homogeneous polynomials. Therefore, research in this field is of great theoretical and practical value. This paper focuses on introducing a novel class of tensors, termed $SD{D}_2$ tensors, which are derived from $SD{D}_2$ matrices and constitute a subclass of strong $\mathcal{H}$-tensors. Furthermore, we also investigate the relationships among $SD{D}_2$ tensors, strong $\mathcal{H}$-tensors, $SD{D}_1$ tensors and $SDD$ tensors. Additionally, we extend the concept of $SD{D}_2$ tensors to $B$-tensors, thereby defining a new tensor class called $B_2$-tensors and analyzing their fundamental properties.

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.