I have the equation $$\underline{x}(t+1) = \textbf{A}\,\underline{x}(t)\, ,\quad (1)$$ where $\underline{x}$ is the $n$-dimensional vector of unknowns and $\textbf{A}$ is a $n\times n$ numerical matrix. I know that $\textbf{A}$ is diagonalizable and that it admits
- an eigenvalue $\lambda_1$ with eigenvector $\underline{x}_1$
- eigenvalues $\lambda_2=\dots=\lambda_{n_1}$ with eigenvectors $\{\underline{x}_2,\dots, \underline{x}_{n_1}\}$
- an eigenvalue $\lambda_{n_1+1}$ with eigenvector $\underline{x}_{n_1+1}$
- eigenvalues $\lambda_{n_1+2}=\dots=\lambda_{n}$ with eigenvectors $\{\underline{x}_{n_1+2},\dots, \underline{x}_{n}\}$
How can I calculate the solution of Eq. (1), $\underline{x}(t)$?