Consider real diagonal (known) matrices $A$, $B$, and $C$, and a comforming matrix $\Pi$.

$C + \Pi B + \Pi A \Pi = 0$

I have been trying to solve this system using elementary algebra. I have two questions: (1) Is this a Ricatti equation? ; and (2) does this have an analytical solution?

Thanks!

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(1) The Nonsymmetric Algebraic Riccati Equation is $$C + \Pi B + D \Pi + \Pi A \Pi = 0$$ The $D$ matrix in your case is zero.
(2) Your situation is much simpler. If each matrix is diagonal, look at $\Pi = diag(x_i)$ as your solution. Then for $A=diag(a_i)$ $B=diag(b_i)$ $C=diag(c_i)$ the matrix equation is easily seperable into the equations: $$c_i + x_ib_i + x_i^2a_i =0$$ And each term is found with the usual quadratic formula. Each equation will have two solutions unless there is a repeated root. So in general, as a system there are $2^n$ different matrices $\Pi$ to solve your formula, the choice of one of two possibilities for each diagonal in $\Pi$.