Consider a linear matrix differential equation of the form
$$\frac{\mathrm{d} C}{\mathrm{d} t} = A C + C A^{\mathrm{T}}$$
where $C$ is a symmetric $n \times n$ matrix and $A$ is a $n \times n$ matrix. Find $C(t)$.
Is there a formal solution for the above equation? This is in principle linear equation if we treat the matrix $C$ and $A$ as a $n^2$ vector. However, it does not seem to be practical way to solve the problem.
This kind of differential equations for matrices is quite new to me. Besides the formal solution let me know some books considering similar topic. Thanks.
You can solve the differential equation as follows: write $$\frac{dC}{dt} = AC+CA^T = \left(A\otimes1+1\otimes A^T\right)C$$ This gives us the solution $$C(t)=\exp\left(t\left(A\otimes1+1\otimes A^T\right)\right)C(0)$$ Notice that this is basically solving by looking at $A$ as a $n^2$-vector, as you said in the question. The nice thing with this formulation is that you notice at once that $A\otimes1$ and $1\otimes A^T$ commute, and thus we have $$\exp\left(t\left(A\otimes1+1\otimes A^T\right)\right) = \exp\left(t\left(A\otimes1\right)\right)\exp\left(t\left(1\otimes A^T\right)\right)=$$ $$=\left(\exp(tA)\otimes1\right)\left(1\otimes \exp\left(tA^T\right)\right)$$ So $$C(t) = \exp(tA)C(0)\exp\left(tA\right)^T$$ which you can easily check to be correct.