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Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
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real symmetric matrix has real eigenvalues - elementary proof
How about Jacobi's proof?
See, e.g., Folkmar Bornemann, ``Teacher's Corner - kurze Beweise mit
langer Wirkung,'' DMV-Mitteilungen 3-2002, Seite 55 (in German, sorry). Sorry, I don't have the original …