Are Eigenvalues of a Non-Hermitian Matrix Real?

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SUMMARY

The discussion centers on the eigenvalues of a 4x4 non-Hermitian matrix and the inquiry into methods for proving their reality without direct computation. It is established that, unlike Hermitian matrices which guarantee real eigenvalues, non-Hermitian matrices typically have complex eigenvalues. The consensus is that there are no general methods to ascertain the reality of eigenvalues for non-Hermitian matrices without performing calculations.

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andrewm
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Given a 4x4 non-Hermitian matrix, is there any method I can use to prove the eigenvalues are real, aside from actually computing them?

I'm looking for something like the converse of the statement "M is Hermitian implies M has real eigenvalues".

When can one say that the eigenvalues of a given matrix are real?
 
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In general, the eigenvalues of a non-Hermitian matrix can be complex. You would need to compute them.
 
OK, I'm not surprised. Thanks anyways.
 

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