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Let $P, Q\in \mathbb{R}^{m\times m}$ are symmetric matrixes.

$A$ is an unknown matrix $\mathbb{R}^{m\times m}$ which satisfies the following equality and $A$ is not sure to be unique, $$A^TAA^TA+A^TAPA^TA-Q=0.$$

Is there any available solver to solve this equation? Thank you in advance.

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First of all redefine $B=A^TA$ and look for symmetric positive definite solution of $B^2+BPB-Q=0$ . This is matrix algebraic Riccati equation, there are algorithms to solve it numerically. –  Alexander Vigodner Jul 3 at 15:14
    
@AlexanderVigodner, thank you very much for your comment. Does every symmetric positive definite matrix $B$ can be decompose to $A^TA$? –  user18481 Jul 3 at 15:30
    
Yes but in infinitely many ways this is why $A$ is not defined uniquely. I am not also sure that $B$ always exists and unique certain conditions for $P$ and $Q$ must be satisfied. –  Alexander Vigodner Jul 3 at 15:32
    
@AlexanderVigodner, Thank you so much. –  user18481 Jul 3 at 15:49

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