# Prove that T has a unique eigenvalue and find the matrix $A =[T]_\beta$

by Juju9704   Last Updated June 12, 2019 07:20 AM

$$V$$ is a vector space over the complex numbers with finite dimension and consider $$T\in L(V)$$ and $$v \in V$$\ {$$0$$} such that $$T^{n-1}v \neq 0$$ but $$T^nv=0 \in V$$

Prove that T has a unique eigenvalue and find the matrix $$A =[T]_\beta$$

where $$\beta=$$ $$v,Tv,T^2v,...,T^{n-1}v$$}

I just know that $$\beta$$ is a set independent, how can I prove what I asked for?

Do I have to use the cyclic vector? How can I find the matrix A?

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