We consider the sequence spaces $s^0_\alpha(\tilde B)$, $s^{(c)}_\alpha(\tilde B)$ and $s_\alpha(\tilde B)$ with their topological properties, and give the characterizations of the classes of matrix transformations from them into any of the spaces $\ell_1$, $\ell_\infty$, $c_0$ and $c$. We also establish some estimates for the norms of bounded linear operators defined by those matrix transformations. Moreover, the Hausdor measure of noncompactness is applied to give necessary and sufficient conditions for a linear operator on the sets $s^0_\alpha(\tilde B)$, $s^{(c)}_\alpha(\tilde B)$ and $s_\alpha(\tilde B)$ to be compact. We also close a gap in the proof of the characterizations by various authors of matrix transformations on matrix domains.