# How to solve the following nonlinear matrix equation?

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.

• Use Kronecker product (math.uwaterloo.ca/~hwolkowi/henry/reports/…) (see in particular p. 11 of this M. Sc. document) Jan 19, 2017 at 15:29
• @JeanMarie Thank you very much for the hint. However, I still don't know how to deal with the quadratic term $XCX$. Jan 19, 2017 at 15:54
• Here is a reference : (citeseerx.ist.psu.edu/viewdoc/…) Jan 19, 2017 at 16:09

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...