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I have seen outlined in this comment os mathoverflow how to solve quadratic matrix equations of the form $$ XCX + AX = I $$ where $X \in \mathbb{R}^{n\times n}$, $C = C^T > 0 \in \mathbb{R}^{n\times n}$, and $I$ is identity of corresponding size. I need to look this up more thoroughly; thus I need a name of a related theorem or a book, paper or lecture to do some advanced reading about conditions for existence and uniqueness of solutions and better even, under what conditions does a positive definite $X$ exist.

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Bini, Iannazzo, Meini, Numerical Solution of algebraic Riccati equations, SIAM books, seems a good starting point to me. It is a monograph that deals both with the symmetric and the non-symmetric case and assumes no previous knowledge in control theory.

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  • $\begingroup$ Thank you. From what I read so far, this seems like a good read and usable for citation. $\endgroup$ – marc Aug 29 '16 at 12:04
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Multiplying by $C$ from the right, the equation is reduced to $$Y^2 + AY - C = O,$$ where $Y=XC$. Solution to such polynomial matrix equations is described in Chapter VIII in F.R.Gantmachers. The Theory Of Matrices.

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