幂等矩阵线性组合的可逆性及逆的表示

Inversibility and Inverse Representation of Linear Combination of Idempotent Matrices

研究一组幂等矩阵在限定条件A_(1)+A_(2)+…+A_(m)=I下表示任一矩阵及其逆矩阵.首先给出一组满足特定条件的对称幂等矩阵的若干性质及其线性组合可逆性的判定,在此基础上讨论任一矩阵可用一组幂等矩阵线性表示的条件,最后给出任一可逆矩阵逆的表示形式.

A set of idempotent matrices are studied to represent any matrix and its inverse under restricted conditions:A_(1)+A_(2)+…+A_(m)=I.Firstly,some properties of a set of symmetric idempotent matrices satisfying certain conditions and the invertibility of linear combinations are given,and then the conditions that any matrix can be linearly represented by a set of idempotent matrices are discussed.Finally,the expression for the inverse of any invertible matrix is given.