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(Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{\top}={?}$

Let me answer the first question: Q: Does $F=\sum_{i=0}^{\infty} A^i (A^i)^{\top}$ for non-symmetric $A$ have a closed-form solution analogous to the solution $\sum_{i=0}^{\infty} A^{2i}=\left( I-A^2 \...
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