Eigenvectors, eigenvalues and matrices

[EugP]

I have:

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

I found that the eigenvalues are r_1 = i and r_2 = - i.
Also, I calculated the eigenvectors to be

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

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

The answer, however, is

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

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

I don't understand what I did wrong.
This is what I did for the eigenvector corresponding to r_1 = i:

\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

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

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

So now I have:

\xi_1 + (-2 - i)\xi_2 = 0

\xi_1 = (2 + i)\xi_2

so:

\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)

Now I let \xi_2 = 1:

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

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

 

[Timo]

If v is an eigenvector, then a*v (with scalar a) is an eigenvector too (by definition and matrices being linear maps). Multiply your solutions with 2-i and 2+i, respectively.

 

[EugP]

If v is an eigenvector, then a*v (with scalar a) is an eigenvector too (by definition and matrices being linear maps). Multiply your solutions with 2-i and 2+i, respectively.

Oh now I see what they did. But my solution is also correct, isn't it?

 

[HallsofIvy]

Yes. The set of all eigenvectors corresponding to a single eigenvalue forms a vector space. There are, necessarily, an infinite number of eigenvectors for any eigenvalue.

 

[EugP]

Yes. The set of all eigenvectors corresponding to a single eigenvalue forms a vector space. There are, necessarily, an infinite number of eigenvectors for any eigenvalue.

Thanks for clearing that up!