幂等矩阵的组合的零度与秩(英文)

THE NULLITY AND RANK OF COMBINATIONS OF IDEMPOTENT MATRICES

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.Rakoei[3]的结果.

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.Rakoceic.