There are well known inequalities for Hermitian matrices A and B that relate the diagonal entries of A+B to the eigenvalues of A and B. These inequalities are easily extended to more general inequalities in the case where the matrices A and B are perturbed through con-gruences of the form UAU*+ V BV *; where U and V are arbitrary unitary matrices, or to sums of more than two matrices. The extremal cases where these inequalities and some generalizations become equal-ities are examined here