Given the matrix $$\hat{S}=\begin{bmatrix} S & *& *&* \\ 0& S &* &* \\ 0& 0& S &* \\ 0&0&0&S\\ \end{bmatrix} $$ where $S$ is an $n \times n$ irreducible matrix, i.e. it can not be equivalent to a block upper-triangular matrix. The stars are arbitrary $n \times n$ matrices in $Com(S)=\{Y; [S,Y]=SY-YS=0 \}$. Required to prove that the above matrix is equivalent to $$\begin{bmatrix} S & I_n& 0&0 \\ 0& S &I_n &0 \\ 0& 0& S &I_n \\ 0&0&0&S\\ \end{bmatrix} $$ Hint for the solution, is the well-known canonical form of nilpotent matrices. The idea is that I try search fro the "well-known canonical form of nilpotent matrices", where using the eigen vectors of a nilpotent matrix , they show that its equivalent to one whose entries are zero except the first off-diagonal composes of ones. My difficulty is how to generalize this result to the case of blocks !!
Thanks for any suggestions