On the eigenvalues of A + B and AB

Abstract

U sing the usual field of values and the angular field of values inclusion sets are found for the eigenvalues of sums and products of n X n complex matrices. For instance if the field of values of B does not contain 0 it is found that the quotient of the field of values of A by that of B contains the eigenvalues of AB -I. Applications are made to the polar form (AB where A is unitary and B positive semidefinite) and to products AB with A hermitian and B + B* positive definite.