We　focus　on　the　convergence　analysis　of　the　extended　linearized　alternating　direction　method　of　multipliers　(L-ADMM)　for　solving　convex　minimization　problems　with　three　or　more　separable　blocks　in　the　objective　functions.　Previous　convergence　analysis　of　the　L-ADMM　needs　to　reduce　the　multi-block　convex　minimization　problems　to　two　blocks　by　grouping　the　variables.　Moreover,　there　has　been　no　rate　of　convergence　analysis　for　the　L-ADMM.　In　this　paper,　we　construct　a　counter　example　to　show　the　failure　of　convergence　of　the　extended　L-ADMM.　We　prove　the　convergence　and　establish　the　sublinear　convergence　rate　of　the　extended　L-ADMM　under　the　assumptions　that　the　proximal　gradient　step　sizes　are　smaller　than　certain　values,　and　any　two　coefficient　matrices　in　linear　constraints　are　orthogonal.　©　2015,　Operations　Research　Society　of　China,　Periodicals　Agency　of　Shanghai　University,　Science　Press　and　Springer-Verlag　Berlin　Heidelberg.