A new class of matrices called partially doubly strictly diagonally dominant (for shortly, PDSDD) matrices is introduced and proved to be a subclass of nonsingular $H$-matrices, which generalizes doubly strictly diagonally dominant matrices. As applications, a new eigenvalue localization set for matrices is given, and an upper bound for the infinity norm bound of the inverse of PDSDD matrices is presented. Based on this bound, a new pseudospectra localization for matrices is derived and a lower bound for distance to instability is obtained.