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[tex]x' = \left(\begin{array}{cc}2&-5\\1&-2\end{array}\right) x[/tex]

I found that the eigenvalues are [tex]r_1 = i[/tex] and [tex]r_2 = - i[/tex].

Also, I calculated the eigenvectors to be

[tex]\xi_1 = \left(\begin{array}{c}2 + i\\1\end{array}\right)[/tex]

[tex]\xi_2 = \left(\begin{array}{c}2 - i\\1\end{array}\right)[/tex]

The answer, however, is

[tex]\xi_1 = \left(\begin{array}{c}5\\2 - i\end{array}\right)[/tex]

[tex]\xi_2 = \left(\begin{array}{c}5\\2 + i\end{array}\right)[/tex]

I don't understand what I did wrong.

This is what I did for the eigenvector corresponding to [tex]r_1 = i[/tex]:

[tex]\left(\begin{array}{cc}2 - i&-5\\1&-2 - i\end{array}\right)\left(\begin{array}{c}\xi_1\\\xi_2\end{array}\right) = 0[/tex]

By multiplying row 2 and subtracting it from row 1, I got:

[tex]\left(\begin{array}{cc}0&0\\1&-2 - i\end{array}\right) \left(\begin{array}{c}\xi_1\\\xi_2\end{array}\right) = 0[/tex]

So now I have:

[tex]\xi_1 + (-2 - i)\xi_2 = 0[/tex]

[tex]\xi_1 = (2 + i)\xi_2[/tex]

so:

[tex]\xi = \left(\begin{array}{c}\xi_1\\\xi_2\end{array}\right) = \left(\begin{array}{c}\(2 + i)\xi_2\\\xi_2\end{array}\right)[/tex]

Now I let [tex]\xi_2 = 1[/tex]:

[tex]\xi = \left(\begin{array}{c}2 + i\\1\end{array}\right)[/tex]

I think I did everything right, but I get the wrong answer. Am I missing something?

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# Eigenvectors, eigenvalues and matrices

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