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

摘要

We study the constrained systemof linear equations Ax=b,x∈R(A^(k))for A∈C^(n×n)and b∈Cn,k=Ind(A).When the system is consistent,it is well known that it has a unique A^(D)b.If the system is inconsistent,then we seek for the least squares solution of the problem and consider min_(x∈R(A^(k)))b−Ax2,where·2 is the 2-norm.For the inconsistent system with a matrix A of index 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.