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Wikipedia says:

Having an eigenvalue is an accidental property of a real matrix (since it may fail to have an eigenvalue), but every complex matrix has an eigenvalue.

Yet, IMO, real matrices are subclass of complex ones. So, even without having any mathematical degree I see that this cannot be true.

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    You can write an answer by yourself, if you want. Regards and happy new year.2012-12-31

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Answer in the comment: The Wikipedia article you linked says it: "Namely, let $V$ be any vector space with some scalar field $K$, and let $T$ be a linear transformation mapping $V$ into $V$. We say that a vector $x$ of $V$ is an eigenvector of $T$ if (and only if) there is a scalar $\lambda\in K$ such that $T(x) = \lambda x$."

($\lambda\in K$) must be in bold

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    An eigenvector should be non-zero too.2012-12-31
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Eigenvalues are roots of a polynomial. Not every real polynomial has real roots. But every complex polynomial has roots.

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    The question is what the polynomial coefficient have to do with the requirement for the roots to be in the same field?2012-12-31