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real symmetric matrix has real eigenvalues - elementary proof

Dear colleagues,

I am looking for short, elementary proof that every real symmetric matrix has at least one real eigenvalue. Elementary means that it is performed on the level of the first semester of the first class, using only matrix algebra (algebraic) methods, using neither complex numbers nor analytic methods (no derivatives, etc.).

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 This looks almost identical to mathoverflow.net/questions/118626/…. – shane.orourke Jan 15 at 10:11
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 this question seems a duplicate of this one mathoverflow.net/questions/118626/… (and the OP has actually contributed to it by answering) – johndoe Jan 15 at 10:16
also, since derivatives are taught in high-school, any particular reason for avoidance of derivatives? – S. Sra Jan 15 at 10:17
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 @Survit: it is even more puzzling to me, why one would want to avoid complex numbers:-) – Alexandre Eremenko Jan 15 at 13:04
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 @Alexandre: I agree, why would anybody want to avoid complex numbers (except when writing code to avoid doubling storage); I am always trying to find excuses to use complex numbers, but not getting enough excuses :-) – S. Sra Jan 15 at 20:44
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closed as exact duplicate by Dan Petersen, Emil Jeřábek, Carlo Beenakker, Alexandre Eremenko, Chris Godsil Jan 15 at 13:36

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