幂等矩阵的组合的零度与秩

本文研究了两个幂等矩阵P与Q的组合Ap Bq-Cpq(a≠0,b≠0)的秩.利用矩阵的核子空间及线性空间的同构的有关性质,得到了:当c＝a b时,Ap Bq-Cpq的秩为一个常数,且等于P-Q的秩;当c≠a b时,Ap Bq-Cpq的秩为一个常数,且等于P Q的秩,推广了J. J. Koliha和V. Rakoeeie3]的结果.

THE NULLITY AND RANK OF COMBINATIONS OF IDEMPOTENT MATRICES

The paper researches the rank of the combinations of two idempotent matrices P and Q, i.e., the rank of aP+bQ-cQP (where a,b,c∈C,a≠0,b≠0). By using the properties of the nullspace of the matrix and isomorphisms of the linearspace, we get that the rank of aP+bQ-cPQ is a constant and is equal to the rank of P-Q when c＝a+b, elsewise equal to the rank of P+Q when c≠a+b, which generalize the results of J. J. Koliha and V. Rakoeeie3].