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  
\[A^n\]
.

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