I am given a system of the form $x'(t)=Ax(t)$ where $A\in M_{3}(\mathbb{R})$ is a diagonalizable matrix and an initial condition $x(0)=\begin{pmatrix}1/3\\ 2/3\\ 0 \end{pmatrix}$ and I am being asked to find the stationary distribution.
How can I find the stationary distribution ?
I have calculated the eigenvales and eigenvectors of $A$ and then solved the ODE by finding a basis for the solution space and choosing a linear combination that would satisfy the initial condition to obtain $x(t)$.
How to I proceed ?