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I want to find a matrix $X$ that satisfies the following matrix equation: $$A+BX=XB+XCX$$ where $A,\ B,\ C$ are all square matrices, and $A,\ C$ are hermitian. Is there a general procedure to solve this kind of matrix equations? Finding a particular solution is enough.

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This Equation is called Algebraic Riccati Equation and is mostly used in control-theory. Often ones uses iterative approcahes to solve this. By starting with an Guess $X_0$ one can find an Error $E$ by setting $$ A+B(X_0 +E) =(X_0+E)B + (X_0+E)C(X_0+E) $$ which is equal to $$ (A+BX_0 -X_0D+X_0CX_0) + (B+X_0C)E = E(B+CX_0) + ECE $$ which is of similar type. Ignoring $ECE$ which is hopefully small, one can solve the resulting equation to get $E$. This results in $X_1 = X_0+E$ and we are back at the start...

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