In my lecture notes it says that for a linear dynamical system of the form $ f(x) = Ax $ where A is diagonalisable d x d matrix, with $ \left \{ v_1 , v_2, \cdots , v_d \right \} $ a basis for $ \mathbb{R^d} $ consisting of eigenvalues of A, then "it is easy to see that" $ \lambda_j^n v_j $ is a particular solution for every $ 1 \leq j \leq d $ . Thus the general solution is $ x_n = \sum_{j=1}^d c_j \lambda_j^n v_j $.
I'm finding it difficult to follow this intuition. I understand that we have $ x_{n+1} = Ax_n = A^2 x_{n-1} = \cdots = A^{n+1}x_0 $ and if A is diagonalisable with $ A = C \Lambda C^{-1} $ then this becomes $ x_n = C\Lambda^n C^{-1}x_0$ with $ \lambda_j^n $ the diagonal values of $ \Lambda^n $, however I don't see how $\lambda_j^n v_j $ is a particular solution.
What is the reasoning here?