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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!


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

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@Theo: Why is this linear algebra? Are matrices always associated with vector spaces? Can they connect to the outside of vector spaces? PS: I identify linear algebra with the theory of vector spaces. – Tim Aug 20 '11 at 14:45
Tim, sometimes you baffle me... – t.b. Aug 20 '11 at 14:46
up vote 7 down vote accepted

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.

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