I have a triangular matrix $A$, and I need to find $A^{20}$.
$$ A = \begin{bmatrix} 3&1&0\\ 0&3&1\\ 0&0&3\\ \end{bmatrix} $$
I tried to use deagonalization method, but can't seem to find the correct eigen values and vectors.
Using $|A-\lambda I|=0$, I get $(3-\lambda)^3 = 0 \implies \lambda = 3$. So I'm guessing the eigen values all are 3?
For eigen vectors, using $|A-\lambda I|x=0$, I get $x_y = -x_z$. I'm not sure what to do next.
This matrix is a Jordan block, it cannot be diagonalized.
For $k \ge 2$, we have $$A^k=\begin{bmatrix} 3^k & k\cdot 3^{k-1} & \binom{k}{2}3^{k-2} \\ 0 & 3^k & k \cdot 3^{k-1} \\ 0 & 0 & 3^k\end{bmatrix}$$
using the formula stated here.