Solve for X where Sum(A.X+C).(A.X + C) = X

Question

I am trying to solve for X (vector of dimension n):

$\Sigma_{i=1}^n(AX+C)_i(AX + C) = X$

where: C = is a constant vector dimension n A = is a diagonal invertible matrix [n,n]

Answer 1

Assume that $A$ is invertible and put $Y=AX+C$. Then $(\sum_iY_i)Y=A^{-1}(Y-C)$. This a system of $n$ equations of degree $2$ in the $n$ unknowns $(Y_i)_i$. Generically (when $A,C$ are full random matrices or also when $A$ is a diagonal random matrix) it seems to have $n+1$ complex solutions.