Are two matrices having the same characteristic and minimal polynomial always similar?

Solution 1:

Consider matrices in Jordan normal form with the same diagonal entries. The minimal polynomial just tells you the size of the biggest Jordan blocks (for the respective Eigenvalues). Example (for some $a$):

$\begin{pmatrix} a & 1 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix},\begin{pmatrix} a & 1 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 1 \\ 0 & 0 & 0 & a\end{pmatrix}$

Solution 2:

Take the matrices

$$\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix}\,\,,\,\,\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}$$

These two matrices have $\,x^4\,$ as char. polynomial and $\,x^2\,$ as minimal one.

Try a nice exercise: prove that the condition is sufficient if the matrix is $\,n\times n\,\,,\,n\leq 3\,$