link between Normal operator and set of linearly independent vectors.

Question

consider $T:\mathbb{R}^{5}\rightarrow \mathbb{R}^{5}$, $T$ is a Normal operator. prove that if $\exists w\in \mathbb{R}^{5}$ so the set :$$\left \{ w,T\left ( w \right ),T^{2}\left ( w \right ),T^{3}\left ( w \right ),T^{4}\left ( w \right ) \right \}$$ is linearly independent, then $T$ has $5$ different eigenvalues.

Answer 1

You can write the matrix $A$ associated to $T$ in that base:

$T(w)=0 w+1 T(w)+ 0 T^2(w) +0 T^3(w)+0T^4(w)$

$T(T(w))=0 w+0 T(w)+ 1 T^2(w) +0 T^3(w)+0T^4(w)$

$T(T^2(w))=0 w+0 T(w)+ 0 T^2(w) +1 T^3(w)+0T^4(w)$

$T(T^3(w))=0 w+0 T(w)+ 0 T^2(w) +0 T^3(w)+1T^4(w)$

$T(T^4(w))=a w+b T(w)+ c T^2(w) +d T^3(w)+eT^4(w)$

So the matrix is

$A=\left[\begin{matrix}0& 0&0&0&a \\ 1 & 0&0&0&b\\0&1&0&0&c\\0&0&1&0&d\\0&0&0&1&e\end{matrix}\right]$