I'm having trouble with this problem. Actually I'm having trouble with all of this set of problems (when the eigenvectors are complex). I must not be finding these things correctly, because nothing is matching up with the book. Any help would be awesome.(adsbygoogle = window.adsbygoogle || []).push({});

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

So...

[tex] \left| \begin{array}{cc} 1-\lambda & 2 \\ -5 & -1-\lambda \end{array} \right| = (1-\lambda)(-1-\lambda)+10=0=\lambda^2+9 [/tex]

[tex] \lambda = 3i[/tex]

So then:

[tex] \left( \begin{array}{cc} 1-3i & 2 \\ -5 & -1-3i \end{array} \right) \left(\begin{array}{cc}x1\\x2 \end{array}\right) =\vec 0 [/tex]

Which is the NULLSPACE of the [tex] \vec A - r\vec I [/tex] matrix.

Gaussian-Jordan reduction yields:

[tex] \left( \begin{array}{cc} 1 & \frac{1}{5}+\frac{3}{5}i \\ 0 & 0 \end{array} \right) [/tex]

[tex] x1=-\left(\frac{1}{5}+\frac{3}{5}i\right)\alpha [/tex]

[tex] x2=\alpha [/tex]

[tex] \vec V = \left( \begin{array}{cc} \frac{-1}{5}-\frac{3}{5}i \\1 \end{array} \right) [/tex]

[tex] \vec V = \left( \begin{array}{cc} \frac{-1}{5} \\ 1 \end{array} \right) + i \left( \begin{array}{cc} \frac{-3}{5} \\ 0 \end{array} \right) [/tex]

So, I'm pretty sure I'm making the mistake in here. But where? Isn't everything I'm doing legit?

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# Homework Help: DIFFEQ - Complex eigenvectors

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