Minimal polynomial of a $4\times4$ matrix

Question

I just need to see an example of a non-diagonalizable $4\times4$ matrix over $\mathbb{R}$ whose minimal polynomial is the same as its characteristic polynomial.

I saw the question elsewhere and became much interested, but I am unable to construct such matrices.

Answer 1

Consider $\begin{pmatrix}\lambda& 1&0&0\\ 0&\lambda &1 &0\\ 0&0&\lambda& 1\\ 0&0&0&\lambda\end{pmatrix}$.The characteristic polynomial and the minimal polynomial are $(x-\lambda)^4$.