I am trying to solve the following matrix equation $(I-X^{-1}BX^{-1})(X-C)=0$ where $X,B,C$ are positive definite and $X$ is unknown. Two obvious solutions are $X=C$ and $X=B^{\frac{1}{2}}$. We also assume that $BC=CB$.
I was wondering whether the other solutions to the above can be computed analytically?
In the specific case $BC = CB$, it is easy to find analytic solutions. Since $B$ and $C$ commute, there is a matrix $P$ such that $P^{-1}BP= D_B$ and $P^{-1}CP = D_C$, where $D_B$ and $D_C$ are diagonal matrices. So, if we choose $X$ to satisfy $X = PD_XP^{-1}$ for some diagonal matrix $D_X$, the equation reduces to $$ [I - (P D_X^{-1}P^{-1})(P D_B^{-1}P^{-1})(P D_X^{-1}P^{-1})][(P D_X^{-1}P^{-1}) - (P D_C^{-1}P^{-1})] = P(I - D_X^{-1}D_BD_X^{-1})(D_X - D_C)P^{-1} = 0 $$ All diagonal matrices commute with each other, so this can be further simplified to $$ (D_X^2 - D_B)(D_X - D_C) = 0 $$ Now, letting $\lambda_{Bi}$ be the entries on the diagonal of $D_B$, and similar for $D_C$ and $D_X$, that equation will be satisfied if, for all $i$, $\lambda_{Xi}^2 = \lambda_{Bi}$ or $\lambda_{Xi} = \lambda_{Ci}$. By choosing one of those two for each $i$, we can create $2^n$ (not necessarily distinct) solutions $X = P^{-1}D_XP$, where $n$ is the size of the matrices.