The minimal polynomial of a transformation

by ayushkhaitan3437

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