Abstract: Let Q=\left(\beginarraycccc-\infty & b_1 & b_2 & \cdots \\ & -q_1 & & \\ & & -q_2 & \\ & & & \ddots\endarray\right) where 0≤b_i<∞,0≤q_i<∞,i≥1. In the present paper, some necessary and sufficient conditions for Q to be a Q-matrix or an honest Q-matrix are given respectively. Besides, it is proved that if Q is a Q-matrix, then there exist infinitely many Q-processes. Similar results for the diagonal Q-matrix with finite number of instantaneous states are derived. The processes constructed here can be explioitly represented by their Laplace transforms.