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We all know that eigenvalues of a Hermitian matrix are real, but I am looking for sufficient conditions for a general real matrix (not necessarily symmetric) to have only real eigenvalues. So far, I know that totally positive matrices have this property I want. Are there any other sufficient conditions besides total positivity?

serige
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    This may be useful:

    http://math.stackexchange.com/questions/669499/what-properties-should-a-matrix-have-if-all-its-eigenvalues-are-real

    From it, $A$ has real eigenvalues $\iff A\sim U$, where $U$ is an upper triangular matrix with real diagonal entries, and $\sim$ is the standard $A\sim B\iff A=PBP^{-1}$ for $P$ invertible.

     – Mark Schultz-Wu Aug 03 '16 at 23:52
  • This MO question is also related, although it doesn't really address your question. – user1551 Aug 04 '16 at 07:58

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