### The minimal polynomial of a transformation

For a linear transformation $T$, the minimal polynomial $m(x)$ is a very interesting polynomial indeed. I will discuss its most interesting property below:

Let $m(x)=p_1(x)^{e_1}p_2(x)^{e_2}\cdots p_s(x)^{e_s}$ be the minimal polynomial of transformation $T$. Then $p_i(T)^{y}\neq 0$ for $1\leq y\leq e_i$ and $y\in\Bbb{N}$. You may be shocked (I hope Mathematics has that kind of effect on you :P). Why this is possible is that the ring of $n\times n$ matrices can have zero-divisors. For example $\begin{pmatrix} 1&1\\2&2\end{pmatrix}\begin{pmatrix}1&1\\-1&-1\end{pmatrix}=0_v$