6
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ @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. $\endgroup$ – Tim Aug 20 '11 at 14:45
  • 11
    $\begingroup$ Tim, sometimes you baffle me... $\endgroup$ – t.b. Aug 20 '11 at 14:46
8
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.