There are $k$ matrix equations with the same unknown $\mathbf{X}$:
$\mathbf{A}_i(\mathbf{D}_i-\mathbf{X})^{-1}\mathbf{B}_i=\mathbf{C}_i$
where $i=1,2,...,k$. $\mathbf{A}_i$ is a $m\times n$ matrix. $\mathbf{D}_i$ and $\mathbf{X}$ are $n\times n$ matrices. $\mathbf{B}_i$ is a $n\times m$ matrix. $\mathbf{C}_i$ is a $m\times m$ matrix.
I have $m<n$ which indicates that each matrix euqation alone is underdetermined. According to [Penrose, 1955], and assuming that the inverses exist, we have the general solution for each equation
$(\mathbf{D}_i-\mathbf{X})^{-1}=\mathbf{A}{_i}^{\dagger}\mathbf{C}_i\mathbf{B}{_i}^{\dagger}+\mathbf{Y}_i-\mathbf{A}{_i}^{\dagger}\mathbf{A}{_i}\mathbf{Y}_i\mathbf{B}{_i}\mathbf{B}{_i}^{\dagger}$
$\Rightarrow\mathbf{X}=\mathbf{D}{_i}-(\mathbf{A}{_i}^{\dagger}\mathbf{C}_i\mathbf{B}{_i}^{\dagger}+\mathbf{Y}_i-\mathbf{A}{_i}^{\dagger}\mathbf{A}{_i}\mathbf{Y}_i\mathbf{B}{_i}\mathbf{B}{_i}^{\dagger})^{-1}$
where $\mathbf{Y}_i$ is an arbitrary matrix with the proper dimensions and $^{\dagger}$ denotes the Moore–Penrose pseudoinverse.
I sucessfully found some $\mathbf{X}$ to satisfy or best satisfy all the $k$ equations, using an optimizer. However, the obtained solution may show an undesired physical meaning. I wonder if this could be done in an analytical way? I think the analytical method is likely to provide a more meaningful result.
Thanks very much for taking time to read this :)