In this paper we are concerned
with analyzing the cases of equality in certain inequalities that relate the eigenvalues
and main diagonal elements of hermitian matrices.
Let E_{r} denote the rth elementary symmetric function of k variables (E_{0} = 1). If
H = (h_{τj}) is an nsquare positive semidefinite hermitian matrix with eigenvalues
γ_{1} ≦⋯ ≦ γ_{n} and if 1 ≦ r ≦ k ≦ n, then it is known that
 (1.1) 
If r > 1 and at least r of h_{11},⋯,h_{kk} are positive then (1.1) can be equality if and only
if there exists a permutation φ ∈_{l}S_{k} such that
 (1.2) 
where H_{n−k} is (n−k)square and + denotes direct sum. Of course, if r = k = n then
(1.1) is the Hadamard determinant theorem:
 (1.3) 
If some h_{ii} = 0, then H is singular and (1.3) is equality. If h_{ii} > 0,i = 1,⋯,n, then
the condition (1.2) yields the wellknown criterion for equality in (1.3), namely
H = diag(h_{11},⋯,h_{nn}).
