幂等矩阵与秩幂等矩阵的充要条件

The Necessary and Sufficient Conditions for Idempotent Matrix and Rank-idempotent Matrix

满足A2=A的n阶方阵A称为幂等矩阵,它是矩阵环Mn(F)的一个幂等元;满足r(A)=r(A2)的n阶方阵A称为秩幂等矩阵。它们与空间的分解、不变子空间的研究有密切关系。利用线性空间的理论方法研究幂等矩阵与秩幂等矩阵的性质,分别得到与它们等价的一些充要条件。

A matrix of satisfying A= A2 is called idempotent matrix.It is an idempotent element of matrix ring Mn（F）.A matrix of satisfying r（A）= r（A2） is called rank-idempotent matrix.They are closely related to decomposition of space and invariant subspace.Some properties of idempotent matrix and rank-idempotent matrix are discussed by using methods of linear space.The necessary and sufficient conditions for idempotent matrix and rank-idempotent matrix are given.