Let S be a completely regular semigroup, and E(S) the partial subgroupoid of idempotents of S. Let γ be a relation on E(S). If γ is a congruence on E(S), that is, if γ is an equivalence relation on E(S) and if χγｙ and μγν satisfy χμγｙν (if both χμ and ｙν are defined in E(S)), then S is called a CS-matrix. Firstly, several characterizations of a CS-matrix are given. Secondly, split CS-matrices are investigated. In particular, matrix representations of these semrgroups are discussed.