"Remarks on the eigenvalues and eigenvectors of Hyper-Hermitian matrices".
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.