Common Solutions to the Matrix Equations $$AX=B$$ A X = B and $$XC=D$$ X C = D on a Subspace

Let $$ \mathbb{S}\mathbb{R}_{{\Omega }}^{n \times n}$$ S R Ω n × n be the set of all $$n \times n$$ n × n symmetric matrices on subspace $${\Omega }$$ Ω , where $$\begin{aligned} {\Omega }=\{ z \in {\mathbb {R}}{^n}|Gz=0,\,G\in {\mathbb {R}}^{k \times n}\}. \end{aligned}$$ Ω = { z ∈ R n | G z = 0 , G ∈ R k × n } . The necessary and sufficient conditions for the matrix equations $$AX=B$$ A X = B and $$XC=D$$ X C = D to have a common solution in $$\mathbb{S}\mathbb{R}_{{\Omega }}^{n \times n}$$ S R Ω n × n and also an expression for the general common solution are obtained. Further, the associated optimal approximate problem to a given matrix $${\tilde{X}} \in {\mathbb {R}}^{n\times n}$$ X ~ ∈ R n × n is discussed and the optimal approximate solution is elucidated. Finally, a numerical experiment is presented to validate the accuracy of our result.