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?

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    $\begingroup$ This may be useful: math.stackexchange.com/questions/669499/… 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. $\endgroup$ – Mark Aug 3 '16 at 23:52
  • $\begingroup$ This MO question is also related, although it doesn't really address your question. $\endgroup$ – user1551 Aug 4 '16 at 7:58

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