Can anyone help me out with the following question:
For the matrix $A$ give a diagonalizable matrix $D$ and a nilpotent matrix $N$ so that $A=D+N$ and $ND=DN$.
$\begin{bmatrix} 1 & 4 \\ -1 & 5 \\ \end{bmatrix} $
I started with finding the Jordan normal form and I got: $\begin{bmatrix} 3 & 1 \\ 0 & 3\\ \end{bmatrix}$ So than J=D+N with D the diagonal matrix and $N=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$. and $A=PJP^{-1}= \begin{bmatrix} 10 & -1\\ 5 & 2\\ \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 0 & 3 \\ \end{bmatrix}*P ^{-1}$ But what are the matrices so that $A=D+N$? And how to compute $A^n$ for $n=1,2,3,\dots$ after that?