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.

使用core-EP逆，我们得到欧几里得范式中约束矩阵最小化问题的唯一解：\（\ mathrm {Minimize} \ \ Vert Mx-b \ Vert _2 \），受约束\（x \在\ mathcal {R}（M ^ k），\）中，其中\（M \ in {\ mathbb {C}} ^ {n \ times n} \），\（k = \ mathrm {Ind}（M）\ ）和\（b \ in {\ mathbb {C}} ^ n \）。对于索引为1的复杂矩阵和非奇异复杂矩阵，此问题可简化为众所周知的结果。我们提出了两种Cramer规则来找到上述问题的唯一解决方案，并为core-EP逆应用了一个众所周知的表达式和一个新表达式。此外，我们考虑了基于W加权核EP逆的相应约束矩阵逼近问题及其Cramer规则。提出了与经典策略的数值比较，用于求解线性等式约束的最小二乘问题。考虑了所考虑的约束优化问题的特殊情况以及在求解约束矩阵方程中的应用。