## Minimum Polynomial of a Matrix That Sends ith Basis Elemts Onto i+1th Element and the nth Element to Zero

Suppose a linear transformation
$T$
sends standard basis elements onto other standard basis element as follows:
$T(\mathbf{e}_j) = \left\{ \begin{array}{11} \mathbf{e}_{j+1} & j=0,...,n-1 \\ 0 & j=n \end{array} \right.$

$T(\mathbf{e}_1)=\mathbf{e}_2, \: T(\mathbf{e}_2)=\mathbf{e}_3 , \: ..., T(\mathbf{e}_{n-1})=\mathbf{e}_n , \: T(\mathbf{e}_n) =0$

Hence
$T^n(\mathbf{e}_1)=0$

If
$A$
is the matrix representing
$T$
then
$A^n(\mathbf{e}_1)=0$

$A$
satisfies a polynomial of degree
$n$
but cannot satisfy a polynomial of degree less than
$n$
since
$T^k(\mathbf{e}_1)=\mathbf{e}_{k+1}$
for
$k . The minimum polynomial is \[A^n$
.