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Let $ A=\begin{bmatrix} -7 & -1 \\ 12 & 0 \\ \end{bmatrix} $ . Find a matrix $ P $ and a diagonal matrix $D$ such that $PDP^{-1} = A$.

Ok so the first thing I need to look for are my eigenvalues and eigenvectors. However, I think I'm doing it wrong and don't seem to get the correct eigenvectors.

If $ det(A-\lambda I)X=0 $ then $ det(\begin{bmatrix} -7-\lambda & -1 \\ 12 & -\lambda \\ \end{bmatrix})= \lambda^2+7\lambda+12=0 $

This gives me eigenvalues $\lambda_1=-3 $ and $\lambda_2 = -4 $


Case: $ \lambda_1 = -3$

$ \begin{bmatrix} -7-(-3) & -1 \\ 12 & -(-3) \\ \end{bmatrix} = \begin{bmatrix} -4 & -1 \\ 12 & 3 \\ \end{bmatrix} $ and end up with eigenvector $\begin{bmatrix} -1/4 \\ 1 \\ \end{bmatrix} $

But this is wrong, and not sure where I made the error, or if my whole procedure is off. So I stopped here.

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    $\begingroup$ There's nothing wrong with what you've done so far. Keep in mind that there are infinitely many possible eigenvectors. $\endgroup$ – bradhd Apr 6 '13 at 0:23
  • $\begingroup$ If I do the same for my second Eigenvalue, I'll get another eigenvector. Now, if I join both eigenvectors, what is that? Is that my $P$ matrix? $\endgroup$ – Dimitri Apr 6 '13 at 0:24
  • $\begingroup$ Yes, that's right. $\endgroup$ – bradhd Apr 6 '13 at 0:25
  • $\begingroup$ Hmmm.. According to this assignment, $ P=\begin{bmatrix} -1/4 & -1/3 \\ 1 & 1 \\ \end{bmatrix} $ is wrong...Not sure why. $\endgroup$ – Dimitri Apr 6 '13 at 0:28
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You can write the solution as:

$$ A= P \cdot D \cdot P^{-1} = \begin{bmatrix} -1 & -1\\3 & 4\end{bmatrix} \cdot \begin{bmatrix} -4 & 0\\0 & -3 \end{bmatrix} \cdot \begin{bmatrix} -4 & -1\\3 & 1\end{bmatrix}$$

Compare the eigenvalues\eigenvectors I wrote and the ones you wrote. So your answer is exactly the same as mine! Look at them closely and ask what you can change to make them look identical.

I would rather stay away from the fractions as they make my math more error prone (just a choice).

Does that make sense?

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  • $\begingroup$ That actually worked. The assignment did not like having the eigenvector in a fraction form...Thank you. $\endgroup$ – Dimitri Apr 6 '13 at 0:46
  • $\begingroup$ @Dimitri: You are very welcome and you are doing great! When you write something to a negative power, you enclose it using ^ and squiggly brackets. Regards $\endgroup$ – Amzoti Apr 6 '13 at 0:47
  • $\begingroup$ OHH! Great, I was trying to figure out how to do that and I gave up! But for sure, I'll make sure not to leave stuff in fraction form. $\endgroup$ – Dimitri Apr 6 '13 at 0:48
  • $\begingroup$ Thanks for the up-votes! I don't so much care about the points, just ways to improve my writing of answers so they are clear and helpful! $\endgroup$ – Amzoti Apr 6 '13 at 1:21
  • $\begingroup$ I was the upvote and wanted to ask why the downvote - no explanation! $\endgroup$ – Amzoti Apr 6 '13 at 2:43

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