On the eigenvalues of A + B and AB
Helmut Wielandt · Journal of Research of the National Bureau of Standards Section B Mathematical Sciences · 1973
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.