This paper contains many
inter-related results dealing with the general question of determination of real
eigenvalues of complex matrices. We first disouss the relationship between the
number of elementary divisors associated with real eigenvalues of a matrix A and the
signature of a Hermitian matrix H when AH is also Hermitian. We then obtain sets
of equivalent conditions for a matrix to be similar to a real matrix; for a matrix to be
symmetrizable; and for a matrix to be similar to a real diagonal matrix. As
corollaries we obtain results on the eigenvalues and elementary divisors of
products of two Hermitian matrices. Some of the results are not new; these are
included to give a more complete survey of what is known in these particular
areas.
