Let $\mathbf{A}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible, symmetric matrix.
Let $\mathbf{Q}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible matrix.
Given these properties of $\mathbf{A}$ and $\mathbf{Q}$, I am interested in whether there exist an analytic form for $\mathbf{X}$ $\in \mathbb{R}^{N \times N}$, the solution to the Lyapunov equation:
\begin{align*} \mathbf{X} \ = \ \int_{0}^\infty \mathrm{e}^{\mathbf{A}t} \ \mathbf{Q} \ \mathrm{e}^{\mathbf{A}t}\, dt \end{align*}
Which is similar to this previously asked stackexchange question, except for that $\mathbf{A}$ is now symmetric, but the two matrix exponential terms are separated by a matrix $\mathbf{Q}$ (which is real and invertible, but not necessarily symmetric). Assume that in general, the matrices $\mathbf{Q}$ and $\mathrm{e}^{\mathbf{A}t}$ do not commute.
Clearly, when $\mathbf{Q}$ $=$ $\mathbf{I}$, the solution is trivial to solve using the rules for integrating a single matrix exponential.
The reason I am interested in solving for $\mathbf{X}$ is that it is the solution to the continuous time Lyapunov equation, and my ultimate goal is to find either an analytic expression to $\mathbf{X}$ within the linear system $(\mathbf{A} \bigotimes \mathbf{I} + \mathbf{I} \bigotimes \mathbf{A})$ vec($\mathbf{X}$) $=$ vec($\mathbf{Q}$), or an analytic expression to $(\mathbf{A} \bigotimes \mathbf{I} + \mathbf{I} \bigotimes \mathbf{A})^{-1}$, exploiting the fact that $\mathbf{A}$ is symmetric, real, and invertible.
This older post discusses it as the solution to the Lyapunov equation, but they are unable to obtain a nice analytic form since they don't assume anything special with the matrices involved. I am hoping that the solution will simplify with my assumptions, e.g. symmetric $\mathbf{A}$.
EDIT:
Something I also noted is that my problem only requires me to determine $\mathbf{X} + \mathbf{X}^\top$. Can I say that $\mathbf{X} + \mathbf{X}^\top$ follows
\begin{align*} \mathbf{X} + \mathbf{X}^\top \ = \ \int_{0}^\infty \mathrm{e}^{\mathbf{A}t} \ (\mathbf{Q} + \mathbf{Q}^\top) \ \mathrm{e}^{\mathbf{A}t}\, dt \end{align*}
Or otherwise exploit that $\mathbf{X} + \mathbf{X}^\top$ is symmetric? Or do further simplifications strictly require some commutation relation between $\mathbf{A}$ and $\mathbf{Q}$ (as implied by Abezhiko)?
