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