The Core-EP,Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations

• 来源期刊: 数学研究通讯
• 年，卷（期）: 2021,（1）
• 页码范围: 86-112
• 页数: 27页
• 语种: 英文
• DOI: 10.4208/cmr.2020-0028

摘要

We study the constrained system of linear equations Ax = b,x∈R(Ak)for A ∈ Cn×n and b € Cn,k = Ind(A).When the system is consistent,it is well known that it has a unique ADb.If the system is inconsistent,then we seek for the least squares solution of the problem and consider min、||b-Ax||2,x∈R(Ak)where ||?‖2 is the 2-norm.For the inconsistent system with a matrix A of in-dex one,it was proved recently that the solution is A(#)b using the core inverse A(#)of A.For matrices of an arbitrary index and an arbitrary b,we show that the solution of the constrained system can be expressed as A(#)b where A(#)is the core-EP inverse of A.We establish two Cramer's rules for the inconsistent constrained least squares solution and develop several explicit expressions for the core-EP inverse of matrices of an arbitrary index.Using these expressions,two Cramer's rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper.We also consider the W-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.