可交换幂等矩阵的性质及推广

The Properties and Extensions of Commutative Idempotent Matrix

幂等矩阵以及它们线性组合的性质在矩阵理论和概率统计中都有重要的应用。在满足AB=BA的条件下分别给出当A为幂等矩阵,B为任意方阵时,线性组合k1A+k2B为幂等矩阵的充分必要条件,并且利用该结果直接得出当A、B均为幂等矩阵时,A与B的和、差、积仍为幂等矩阵的条件;A与B的和、差、积的值域、核,分别与A,B的值域、核之间的关系;当A为幂等矩阵,B为任意方阵时,A的值域与核分别是B的不变子空间的充分必要条件。

It is important to study idempotent matrix and the linear combination of them in matrix theory and statistics. Under the condition of,let A be an idempotent matrix and B be an arbitrary complex matrix,then the sufficient and necessary conditions are given in which the linear combination is an idempotent matrix.And the results are directly used to obtain the following conclusions:When A and B are idempotent matrixes,the conditions of the sum,difference,product of A and B are still as idempotent matrix;The relationship of the image of linear transformation and kernel of A and B's sum,difference,product are respective with the image of linear transformation and kernel of A and B.Let A be an idempotent matrix and B be an arbitrary complex matrix,the sufficient and necessary conditions are given in which the image of linear transformation is obtained and kernel of A is invariant subspace of B.