I have some issues in finding an asymptotic expansion for a square root matrix and I have already posted a question (Asymptotic expansion square root matrix). Somebody redirected me to a post where there is a possible answer to my question (How to calculate the square root of matrix $A+B$ perturbatively?). However I am not totally sure about such an answer and I would like to share my doubts.

Let formalize a little bit the definition of quantities we are dealing with in the latter post. The quantity we focused on may be thought to have an expansion in powers of $n^{-1/2}$ of the form $A+n^{-1/2}B+O(n^{-1})$ as $n\rightarrow\infty$, where $A=O(1)$ and $B=O(1)$ are symmetric positive definite matrices. Suppose that we want the same expansion for the square root matrix. Such a matrix is assumed to be symmetric (and obviously positive definite) as well and to have the expansion $C+n^{-1/2}D+O(n^{-1})$, with $C=O(1)$ and $D=O(1)$. By applying the identity $$(C+n^{-1/2}D)(C+n^{-1/2}D)=CC+n^{-1/2}CD+n^{-1/2}DC+O(n^{-1})=A+n^{-1/2}B+O(n^{-1})$$ and by matching the expansion term by term we are able to find an expansion for the square root matrix in terms of $A$ and $B$. In particular, $C$ and $D$ must fulfill the equations $A=CC$ and $B=CD+DC$. I hope that all of us would agree that $C=A^{1/2}$, so my point is the following: how it is possible that the solution of the second equation $B=A^{1/2}D+DA^{1/2}$ is $D=A^{-1/2}B/2$? I was told that this is a Lyapunov equation and for this kind of equation, I have not seen such a beauty and explicit solution in G. R. Duan and R. J. Patton (1998). Explicit and analytical solutions to Lyapunov algebraic matrix equations. UKACC International Conference on CONTROL 1998.

Any ideas?

I have some issues in finding an asymptotic expansion for a square root matrix and I have already posted a question (Asymptotic expansion square root matrix). Somebody redirected me to a post where there is a possible answer to my question (How to calculate the square root of matrix $A+B$ perturbatively?). However I am not totally sure about such an answer and I would like to share my doubts.

Let formalize a little bit the definition of quantities we are dealing with in the latter post. The quantity we focused on may be thought to have an expansion in powers of $n^{-1/2}$ of the form $A+n^{-1/2}B+O(n^{-1})$ as $n\rightarrow\infty$, where $A=O(1)$ and $B=O(1)$ are symmetric positive definite matrices. Suppose that we want the same expansion for the square root matrix. Such a matrix is assumed to be symmetric (and obviously positive definite) as well and to have the expansion $C+n^{-1/2}D+O(n^{-1})$, with $C=O(1)$ and $D=O(1)$. By applying the identity $$(C+n^{-1/2}D)(C+n^{-1/2}D)=CC+n^{-1/2}CD+n^{-1/2}DC+O(n^{-1})=A+n^{-1/2}B+O(n^{-1})$$ and by matching the expansion term by term we are able to find an expansion for the square root matrix in terms of $A$ and $B$. In particular, $C$ and $D$ must fulfill the equations $A=CC$ and $B=CD+DC$. I hope that all of us would agree that $C=A^{1/2}$, so my point is the following: how it is possible that the solution of the second equation $B=A^{1/2}D+DA^{1/2}$ is $D=A^{-1/2}B/2$? I was told that this is a Lyapunov equation and for this kind of equation, I have not seen such a beauty and explicit solution in G. R. Duan and R. J. Patton (1998). Explicit and analytical solutions to Lyapunov algebraic matrix equations. UKACC International Conference on CONTROL 1998.

Any ideas?