On SDD$_1$ matrices

The paper continues the study of the recently introduced class of SDD$_1$ matrices. The class of general SDD$_1$ matrices and three its subclasses are considered. In particular, it is shown that SDD$_1$ matrices are nonsingular $\mathcal{H}$-matrices. Also parameter-free upper bounds for the $l_\infty$-norm of the inverses to SDD$_1$ matrices are derived. The block triangular form to which any SDD$_1$ matrix can be brought by a symmetric permutation of its rows and columns is described.