交换环上幂等矩阵的对角化和秩

Diagonability and Ranks of Idempotent Matrices Over Commutative Rings

采用局部化的方法,讨论了交换环上幂等矩阵的对角化问题.以此为工具,得到了交换环上幂等矩阵的秩和它的迹相等的条件,同时得到了交换环上对合矩阵的相关结果.结果表明,对于强联通环上的幂等矩阵,它的秩和它的迹必然相等,从而推广了域上幂等矩阵的相应结果.

Localization method is adopted and the problem of diagonability of idempotent matrices is studied. Based on it, we have got the conditions under which the rank of idempotent matrices equals to its trace over commutative rings and at the same time the rank of involutory ma- trices equals to its trace too. The results show that the rank of idempotent matrices over strongly connected rings will definitely equal to its trace. Thus the results promote the corresponding results of idempotent matrices over field.