A new subclass of nonsingular $H$-matrix named $SD{D}_1^{+}$ matrices is studied in this paper. The relationships between $SD{D}_1^{+}$ matrices and other subclasses of nonsingular $H$-matrices are analyzed by numerical examples. Moreover, the infinity norm bounds of the inverse for $SD{D}_1^{+}$ matrices are derived in two different methods. As applications, two error bounds of the linear complementarity problems ( $LCP$) for $SD{D}_1^{+}$ matrices are given. Finally, the effectiveness of corresponding results is illustrated by numerical examples.