Let $T:\mathbb{R}^2\to \mathbb{R}^2 $ be a linear transformation with characteristic polynomial $x^2+2x-3$. Find the characteristic polynomial of $T^3$.

How to do this?


  • 2
    $\begingroup$ I'm guessing you mean $T: \mathbb{R}^2 \to \mathbb{R}^2$. Do you know the relation between the eigenvalues of $T$ and $T^3$? $\endgroup$ – Pedro M. Jul 26 '13 at 13:20
  • $\begingroup$ Yes, You are correct! $\endgroup$ – 17SI.34SA Jul 26 '13 at 13:23

HINT: If $\lambda$ is an eigenvalue of $\operatorname{T}$ then $\lambda^n$ is an eigenvalue of $\operatorname{T}^n$.


Note that the transformation matrix of $T$ is symmetric with$\lambda=1,-3$. Hence the eigenvalues of the transformation matrix of $T^3$ are $1,-27$. So the characteristic equation is $$(x-1)(x+27)=0$$


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