This　paper　investigates　the　leader-following　consensus　problem　of　multiple　single-integrators　by　using　a　novel　linear　transformation　method　together　with　the　input-to-state　stability　property.　The　multi-agent　system　is　assumed　to　contain　a　single　leader,　and　we　consider　the　following　three　cases.　1)　The　leaders　input　is　pre-given　and　known　by　all　following　agents.　2)　The　leaders　input　is　unknown.　3)　The　leaders　input　is　measurable　online　and　transmitted　to　some　of　follower　agents.　By　constructing　a　transformation　matrix　based　on　incidence　matrix　of　a　virtual　leader-rooted　spanning　tree,　we　make　an　equivalent　transformation　from　leader-following　consensus　problem　to　an　input-to-state　stability　problem.　Then　we　give　a　necessary　and　sufficient　condition,　which　is　the　Hurwitz　stability　of　a　matrix　associated　with　the　communication　topology,　for　ensuring　the　leader-following　consensus.　In　order　to　efficiently　check　whether　the　matrix　is　Hurwitz　stable,　especially　for　large-scale　multi-agent　systems,　we　further　employ　the　Hurwitz　stability　criteria　of　the　matrix　based　on　Metzler　matrix　theory.　Finally,　we　give　numerical　examples　to　validate　the　theoretical　results.　©　2021,　The　Editor(s)　(if　applicable)　and　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.