Abstract.
A quaternionic version of Arnold's results on symmetric (the real case) and
Hermitian (complex case) matrices is explained. These results are a part of
the R-C-H Trinity in Mathematics. It is shown that the eigenvectors form
vector bundles with non-trivial Pontryagin numbers. Nontriviality of these
bundles near matrices with multiple eigenvalues provide a hyper-Hermitian
version of the quantum Hall effect. Among other things the definition of the
quaternionic determinant for a hyper-Hermitian matrix is given whose square
is the determinant of the corresponding complex Hermitian operator.