Solvability of some constrained matrix approximation problems using core-EP inverses

Abstract

Using the core-EP inverse, we obtain the unique solution to the constrained matrix minimization problem in the Euclidean norm: $\mathrm{Minimize }\ \Vert Mx-b\Vert _2$, subject to the constraint $x\in \mathcal{R}(M^k),$ where $M\in {\mathbb {C}}^{n\times n}$, $k=\mathrm {Ind}(M)$ and $b\in {\mathbb {C}}^n$. This problem reduces to well-known results for complex matrices of index one and for nonsingular complex matrices. We present two kinds of Cramer’s rules for finding unique solution to the above mentioned problem, applying one well-known expression and one new expression for core-EP inverse. Also, we consider a corresponding constrained matrix approximation problem and its Cramer’s rules based on the W-weighted core-EP inverse. Numerical comparison with classical strategies for solving the least squares problems with linear equality constraints is presented. Particular cases of the considered constrained optimization problem are considered as well as application in solving constrained matrix equations.