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

Jun Ji & Yimin Wei
2021 Communications in Mathematical Research  
We study the constrained system of linear equations for A ∈ C n×n and b ∈ C n , 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 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
more » ... 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 * Corresponding author. 87 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.
doi:10.4208/cmr.2020-0028 fatcat:wuo6lwql35cd3c5xd7l6zdk7wy