0
$\begingroup$

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?

Thanks!

$\endgroup$
  • 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
4
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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 $$(x-1)(x+27)=0$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.