I am trying to solve for K in the following problem:


$ 3I = A\_1 + A\_2 + A\_3$


$ A\_1 K A\_1 = K\_1 $


$ A\_2 K A\_2 = K\_2 $


$ A\_3 K A\_3 = K\_3 $


Where $I$ is the identity, $K, K\_1, K\_2, K\_3, A\_1, A\_2, A\_3$ are known to be symmetric and positive definite. $K, A\_1, A\_2, A\_3$ are unknown. $K\_1, K\_2, K\_3$ are know. 


* I have attempted to shape the problem into a large $A K A = K\_{1,2,3}$ problem where I just have to solve for $K$ but have been unsuccessful.
* I have also tried to solve by substitution, but I always get multiplications among the $A$ matrices that I am unable to decouple.
* The formulation almost resembles that of the [Algebraic Riccati equation](http://www.wikiwand.com/en/Algebraic_Riccati_equation) in Optimal Control (if we could make it look something like $CK\_1C = K\_2$ and solve using Riccati).


Any ideas?



How to solve this system of Matrix equations? (Coupled riccati equations?)
Not sure if I misunderstood something, but the following seems to work. Let us change the unknown positive-definite variables from the tuple $(A\_{1},A\_{2},A\_{3},K) \mapsto (X\_{1},X\_{2},X\_{3},K)$ where $X\_{i} := A\_{i}K^{1/2}$ for $i=1,2,3$ (all positive definite matrices have unique pos. def. square root).


Then the last three equations become $X\_{i}X\_{i}^{\top} = K\_{i}$, which have solutions $X\_{i} = K\_{i}^{1/2}U\_{i}$, $i=1,2,3$, for arbitrary square orthogonal matrices $U\_{1},U\_{2},U\_{3}$. Therefore,


$$\displaystyle\sum\_{i}X\_{i} = 3 K^{1/2} = \displaystyle\sum\_{i} K\_{i}^{1/2}U\_{i} \quad \Rightarrow \quad K = \left(\frac{1}{3}\displaystyle\sum\_{i} K\_{i}^{1/2}U\_{i}\right)^{2}.$$


Back to original unknowns: $A\_{i} = X\_{i}K^{-1/2} = K\_{i}^{1/2}U\_{i} \left(\frac{1}{3}\displaystyle\sum\_{i} K\_{i}^{1/2}U\_{i}\right)^{-1}$.


