Minimal polynomial of a 6x6 matrix? for the matrix $$
        \begin{pmatrix}
        4 & 1 & 0 & 0 & 0 & 0 \\
        -1 & 2 & 0 & 0 & 0 & 0\\
        0 & 0 & -1 & -2 & -2 & 0 \\
        0 & 0 & 0 & 1 & 0 & 0 \\
        0 & 0 & 4 & 4 & 5 & 0 \\
        0 & 0 & 0 & 0 & 0 & 3 \\
        \end{pmatrix}
$$
I found the characteristic polynomial to be $(t-1)^2(t-3)^4$. How do I go about finding the minimal polynomial?
edit: after looking through my notes, I found $(t-1)^2(t-3)^2$ as the minimal polynomial. Is this right?
 A: I know this question is old but the answers already posted are really misleading.
The matrix is a block diagonal matrix. The minimal polynomial of the first block is $(x-3)^2$. The minimal polynomial of the second block is $(x-1)(x-3)$. The minimal polynomial of the third block is $x-3$. Therefore, the minimal polynomial of the big matrix is the least common multiple of these three polynomials, that is:
$$(x-1)(x-3)^2.$$
A: The minimal polynomial must be $(t-1)^r(t-3)^s$ where $1\le r\le 2$
and $1\le s\le 4$. You could try all possibilities.
But if you suspect it is actually $(t-1)^2(t-3)^2$, all you need
to check is that $(M-I)^2(M-3I)^2=0$, but that
$(M-I)(M-3I)^2=0$ and $(M-I)^2(M-3I)$ are both nonzero.
A: You know that the minimal polynomial is a factor of $(t-1)^2(t-3)^4$ by Cayley-Hamilton, so you just need to determine which factors of $(t-1)^2(t-3)^4$ are polynomials  that vanish when you plug in your matrix $M$, ie, you need to find which factors of $p(t)$ satisfy $p(M) = 0$.
Here the factors are of the form $(t-1)^i(t-3)^j$ with $i \le 2, j \le 4$.
For instance, you would see that $(t-1)^2(t-3)^4$ is forced to be the minimal polynomial were $(M-1)(M-3)^4$ and $(M-1)^2(M-3)^3$ were both found to be nonzero.  But if $(M-1)(M-3)^4$ were zero, then the next step would be to check $(M-3)^4$ and $(M-1)(M-3)^3$, and you would proceed similarly until finished.
Alternatively, if you computed the Jordan normal form of your matrix, you would be done.  Here the minimal polynomial would be $(t-1)^i(t-3)^j$ with the exponents $i,j$ to be the sizes of the largest Jordan blocks corresponding to eigenvalues $1,3$, respectively.
