幂等矩阵线性组合表出零矩阵和单位矩阵的研究

The Researches on the Zero Matrices and Identity Matrices for Linear Expression of Idempotent Matrices

研究3个不同的乘积两两可交换的非零幂等矩阵P1,P2和P3的线性组合表出零矩阵或单位矩阵的所有可能的情况.使用了与已有文献不同的方法,从所对应的{0,1}上线性方程组的全非零解的存在性和结构出发,对问题作了完整的解答;特别是当幂等矩阵P1,P2和P3线性无关时,在置换相似下,只有以(1,1,1),(-1,1,1)和12(1,1,1)为组合系数的3种方式来表出单位矩阵.

This paper studied all the situations that the zero matrix and identity can be denoted by the linear combination of three nonzero idempotent matrices, which are not different from each other and the product of any two matrices is commutative. The method is different from previous literatures. Starting from the existence and structure of all the nonzero solutions for the system of linear equations,the complete resolve for the question is proposed. Specially, when the three matrices are linearly independent, under the 1 permutation similarity, the identity is only denoted by the three ways of the combined coefficients with （1,1,1）, （- 1,1,1） and 1/2 （1,1,1）.