On real eigenvalues of complex matrices

Abstract

This paper contains many inter-related results dealing with the general question of determination of real eigenvalues of complex matrices. We first discuss 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,. © 1965 by Pacific Journal of Mathematics.