For a discrete dynamical system I know that the transition matrix A is diagonalizable with the eigenvalues of 0.1, 0.2, and 0.3. The question asks what I can say about the long term behaviour of the system.
I know that $$ X_k = A^{k}X_0 $$ and $$ A = PDP^{-1} $$ $$ A^k = PD^kP^{-1} $$
Thus I choose to approach the question with by combining the two equations to: $$ X_k = PD^kP^{-1}X_0 $$ as I know that D is \begin{pmatrix}0.1&0&&0\\ 0&0.2&&0\\ 0&0&&0.3\end{pmatrix}
Though as 0.1, 0.2, and 0.3 to a large number (k) all equal 0. Thus my equation becomes: $$ X_k = P \begin{pmatrix}0&0&&0\\\ 0&0&&0\\\ 0&0&&0\end{pmatrix} P^{-1}X_0 $$ so I get $$X_k = \begin{pmatrix}0&0&&0\\\ 0&0&&0\\\ 0&0&&0\end{pmatrix} $$
My question is that this seems too simple and the steady state should not be a zero matrix. I feel like I am missing something but the eigenvalues of the diagonalizable transition matrix is the only information we are given and I have no idea what else the answer can be. Is this right or am I missing something?