Some considerations of matrix equations using the concept of reproductivity (Q2853293)

The authors analyze the matrix equation \(A^mXB^n=C\) and the matrix systems \(A^mX=B\land XD^n=E\) and \(AXA=A\land A^kX=XA^k\). They use the notion of \(k\)-commutative \(\{1\}\)-inverses, where \(\{1\}\)-inverse of the matrix \(A\) is the solution of matrix equation \(AXA=A\). Using the concept of reproductivity, the authors prove that certain formulas represent the general solutions of the analyzed matrix equation and the analyzed matrix systems. The authors determine reproductive and non-reproductive general solutions of the analyzed matrix equation and the analyzed matrix systems. Some of the proved results are known, but the authors give completely new proofs and generalization of these results as well.