0
$\begingroup$

Find a basis $\gamma$ with respect to which both of the following lienar transformations on $\mathbb{R^3}$ become diagionalised (the matrices below are the matrices with respect to the standard basis):

$$S=\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$

and $$T=\begin{bmatrix} 2 & -1 & 2 \\ -1 & 2 & 2 \\ 2 & 2 & -1 \\ \end{bmatrix}$$

I found a basis of eigenvectors for S and another one for T but they are not the same, how should I proceed?

$\endgroup$
  • 1
    $\begingroup$ Hint: Does $ST = TS$? $\endgroup$ – Eric Towers May 5 '16 at 13:10
  • 1
    $\begingroup$ Hint: $T$ has distinct eigenvalues, so any eigenvector of $T$ is also an eigenvector of $S$ $\endgroup$ – Omnomnomnom May 5 '16 at 13:40
  • 1
    $\begingroup$ @OOmnomnomnom: Maxima says $T$ has only $2$ eigenvalues, $-3$ and $3$. $\endgroup$ – Bernard May 5 '16 at 13:46
  • 1
    $\begingroup$ 3 is a repeated eigenvalue $\endgroup$ – Lemma May 5 '16 at 14:10
  • $\begingroup$ Could you further you explanation please, I am missing pieces of the jigsaw puzzle $\endgroup$ – Kam May 6 '16 at 12:46
1
$\begingroup$

Hint: Try the transformation matrix (or it's inverse depending on what you call the transformation matrix) $$\begin{bmatrix} 1 & 1 & -2\\ 1 & -1 &-2\\ -2& 0 &-2\\ \end{bmatrix}$$

$\endgroup$

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.