\[f( \lambda )\]
and minimum polynomial \[m( \lambda )\]
of a matrix \[A\]
then we can find the Jordan canonical form of \[A\]
Suppose
\[m( \lambda)= (2- \lambda )^4(5- \lambda), \: f( \lambda)= (2- \lambda )^5(5- \lambda)^3 \]
The Jordan canonical form of the matrix is then
\[ \left( \begin{array}{cccccccc} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 \end{array} \right) \]
.