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.