Showing that $T:V\to V$ has a cyclic vector if its eigenspaces all have dimension one.

Hints:

  • It suffices to show that the minimal polynomial of $m(x)$ of $T$ is equal to the characteristic polynomial $p(x)$ of $T$, which has degree $n$.
  • If $\lambda$ is an eigenvalue of $T$, then the multiplicity of $\lambda$ in $m$ is equal to the size of the largest Jordan block corresponding to $\lambda$.
  • If $\lambda$ is an eigenvalue of $T$, then the algebraic multiplicity of $\lambda$, i.e. the multiplicity of $\lambda$ in $p$, is equal to the sum of the sizes of the Jordan blocks corresponding to $\lambda$.
  • If $\lambda$ is an eigenvalue of $T$, then the number of Jordan blocks corresponding to $\lambda$ is equal to the geometric multiplicity of $\lambda$, which in your problem is $1$ for all eigenvalues $\lambda$.

Tags:

Linear Algebra

Abstract Algebra

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