We deal with the state consensus problem of a general heterogeneous linear multi-agent system under a time-invariant and directed communication topology. First we adopt a general linear consensus protocol consisting of two parts. One is a state feedback of the agent for independently regulating its dynamics, and the other is a cooperative term in a generalized feedback form of the relative states between the agents. Then we propose a state-linear-transformation to equivalently transform the state consensus problem into a partial stability problem. Therefore, the result from the partial stability theory is applied to derive a sufficient and necessary algebraic criterion of consensus convergence, which is expressed in terms of the Hurwitz stability of a real matrix constructed from the parameters of both the agents’ models and the protocol. Meanwhile, an analytical formula of the consensus function is presented. Based on the criterion, we propose a design procedure of the gain matrices in the protocol by solving a bilinear matrix inequality.