The best approximation problems between the least-squares solution manifolds of two matrix equations

In this paper, we will deal with the following two classes of best approximation problems about the linear manifolds: Problem 1. Given matrices $A_1,B_1,C_1,$ and $D_1\in {\mathbb{R}}^{m\times n}$, find $d(L_1,L_2)=\underset{X\in L_1,Y\in L_2}{min}\Vert X-Y\Vert ,$ and find $\hat{X}\in L_1,\hat{Y}\in L_2$ such that $\Vert \hat{X}-\hat{Y}\Vert =d(L_1,L_2)$, where $L_1=\left\{X\in {\mathbb{S}\mathbb{R}}^{n\times n}|\Vert A_{1}X-B_1\Vert =min\right\}$ and $L_2=\left\{Y\in {\mathbb{S}\mathbb{R}}^{n\times n}|\Vert C_{1}Y-D_1\Vert =min\right\}$. Problem 2. Given matrices $A_2,B_2,E_2,F_2\in {\mathbb{R}}^{m\times n}$ and $C_2,D_2,G_2,H_2\in {\mathbb{R}}^{n\times p}$, find $d(L_3,L_4)=\underset{X\in L_3,Y\in L_4}{min}\Vert X-Y\Vert ,$ and find $\tilde{X}\in L_3,\tilde{Y}\in L_4$ such that $\Vert \tilde{X}-\tilde{Y}\Vert =d(L_3,L_4)$, where $L_3=\left\{X\in {\mathbb{R}}^{n\times n}|\Vert A_{2}X-B_2{\Vert }^2+||X{C}_2-D_2{\Vert }^2=min\right\}$ and $L_4=\left\{Y\in {\mathbb{R}}^{n\times n}|\Vert E_{2}Y-F_2{\Vert }^2+||Y{G}_2-H_2{\Vert }^2=min\right\}$. We obtain explicit formulas for $d(L_1,L_2)$ and $d(L_3,L_4),$ and all the matrices in question by using the singular value decompositions and the canonical correlation decompositions of matrices.