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sorry if this is a dumb question, but I was doing some homework and I noticed that whenever I solved a planar system which had complex eigenvalues, I would always end up with complex eigenvectors. I was wondering whether I could ever somehow get a totally real-valued eigenvector from a complex eigenvalue.

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The short answer is no. Let's suppose that $A$ is a real matrix with a complex eigenvalue $\lambda$ and an associated (nonzero) real eigenvector $v$. Then by the definition of eigenvector, $Av = \lambda v$. So $\lambda v$ must be complex since $v$ has real entries but $\lambda$ is not real; on the other hand, $Av$ must be real since both $A$ and $v$ have real entries by assumption.

This is impossible. Now we can say that if $A$ is a real matrix with a complex eigenvalue, then any associated eigenvector cannot have only real entries.

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    $\begingroup$ Surely the short answer to the question 'Do complex eigenvalues for planar systems with a real valued matrix always have complex-valued eigenvectors?' is 'yes', or am I reading your reasoning wrong? $\endgroup$ – Pete Kirkham Dec 9 '19 at 15:45
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    $\begingroup$ I answered the question @Broderick asked in the last sentence. You are reading reasoning is correctly. $\endgroup$ – Lee Fisher Dec 9 '19 at 15:50

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