广义投影矩阵线性组合的研究

Some Results on the Linear Combinations of Generalized Projectors

投影矩阵和广义投影矩阵的线性组合在矩阵理论的研究中具有重要的意义,并广泛应用到统计、计算、优化、控制等学科领域中.假设A和B是两个n×n非零复矩阵,令P为A、B的线性组合,则P可表示为P=c1A+c2B,其中c1、c2为非零复数.在AB=BA的条件下,对A为广义投影、B为任意矩阵时,线性组合P分别为幂等矩阵、三次幂等矩阵、广义投影矩阵和超广义投影矩阵的充分必要条件进行了研究.

Investigation of linear combinations of projector matrices and hypergeneralized projector matrices is an important topic in the matrix theory.It is widly applied to statistics,computing,optimization,control and other disciplines.Let A and B be n × n nonzero complex matrices,P a linear combination of A and B,so that P = c 1 A ＋ c 2 A,where c 1,c 2 are nonzero complex numbers.In this paper,under the condition AB = BA,the sufficient and necessary conditions are given for P to be idempotent,tripotency,generalized projector and hypergeneralized projector,where A is a generalized projector and B an arbitrary matrix.