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In a project, I came to the following matrix problem: $$C_1=wC$$ $$C=I+wCw^\dagger$$ Where the unknown matrices are $C$, which is hermitian positive definite, and $w$, general not hermitian, no special properties matrix, and the parameter is matrix $C_1$, again general matrix. I had spent quite a bit of time tackling this system, but cannot make satisfactory progress. Especially, the last equation is stiffling me since $w$ enters as self as well as hermitian conjugate, and I cannot quite do anything with it. Is the last equation and its properties, solutions something known in linear algebra? It looks general enough to have appeared somewhere way before me. I am a theoretical physicist, so good with math but not a specialist in the fields.

Specifically, I am interested in the solution or at least the uniqueness of a solution for given $C_1$.

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Note that $C=I+C_1\omega^*$; then to solve your system is equivalent to find $\omega$. On the other hand, $C_1=\omega+\omega C_1\omega^*$. In the real case, we obtain $\omega C_1\omega^T+\omega-C_1=0$, that is a very complicated equation. Generally, there are many solutions and I think that you must use a numerical method to solve this type of equation.

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