I have a specific system of quadratic equations which I try to solve analytically. The system consists of $n$ unknowns $x_1 ... x_n$ which are on the diagonal of matrix $X$, three $n \text{ x } n$ coefficient matrices $A$, $B$, $C$ and to make the equations add up by rows I include a unit vector $e$ of dimension $n \text{ x } 1$. The system looks like
$$
A(X-I)(X-I)e + B(X-I)(X+I)e = Ce,
$$
where $I$ is an identity matrix. Since the X matrix of unknowns is diagonal, I hoped to exploit some of the mathematical identities, like $(X-I)(X+I)=XX-I$ but so far without any luck. Do you have some ideas how to solve this?
Let $Y=X-I=diag((y_i))$. Then $(A+B)Y^2e+2BYe=Ce$ and generically $Y^2e=-2(A+B)^{-1}BYe+(A+B)^{-1}Ce$; putting $E=-2(A+B)^{-1}B,v=(A+B)^{-1}Ce=(v_i)$, we obtain $Y^2e=EYe+v$ or $[y_1^2,\cdots,y_n^2]^T=E[y_1,\cdots,y_n]^T+v$, that is $n$ equations of degree $2$ in the $n$ unknowns $(y_i)$. Generically, there are at most $2^n$ solutions and of course, there is no closed form for the solutions.