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The eigenvalues of a symmetric real matrix are all real. I was wondering if there are conditions either more general than symmetry or that may or may not overlap with symmetry to ensure eigenvalues to be real? Thanks!

Motivation:

A real matrix admits a real Schur decomposition if and only if all of its eigenvalues are real.

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    Tim, sometimes you baffle me...2011-08-20

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A totally positive matrix (meaning that all subdeterminants are positive) has positive and simple eigenvalues.

A totally nonnegative matrix (meaning that all subdeterminants are nonnegative) has nonnegative eigenvalues, but not necessary simple.

See Sergey Fomin's minicourse for links to more info.