This　paper　studies　the　state　consensus　problem　of　a　leaderless　second-order　heterogeneous　linear　multi-agent　system　under　a　fixed　directed　communication　topology.　First,　this　paper　proposes　a　generic　consensus　protocol　with　different　gain　matrices　and　communication　weights　in　matrix　form.　Then,　this　paper　adopts　a　state　linear　transformation　based　on　a　virtual　directed　spanning　tree　of　the　communication　topology　to　convert　the　consensus　problem　into　a　partial　variable　stability　problem.　For　the　second-order　systems　of　general　form　and　a　special　form,　this　paper　deduces　sufficient　and　necessary　consensus　criteria　in　terms　of　Hurwitz　stability　of　corresponding　matrices,　respectively.　According　to　these　criteria,　iterative　algorithms　are　presented　to　find　the　gain　matrices　in　the　protocol　by　solving　some　bilinear　matrix　inequalities.　Finally,　the　effectiveness　of　the　proposed　theoretic　results　are　verified　by　providing　simulation　examples　of　leaderless　heterogeneous　multi-agent　systems　in　general　and　special　cases.　©　2021　IEEE