An analysis of equality in certain matrix inequalities. I
William J. Gordon, Marvin D. Marcus · Pacific Journal of Mathematics · 1970
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 r th elementary symmetric function of k variables (E o = 1).If H=(hij) is an w-square positive semidefinite hermitian matrix with eigenvalues γ^ ••• ^γn and if 1 ^ r ^ k ^ n, then it is known that (1.1) Er(hn, , ft**) ^ Er(n, ''', ϊk) .If r > 1 and at least r of fen, , h kk are positive then (1.1) can be equality if and only if there exists a permutation ce S k such that (1.2) H = diag (χ φa) , , 7v) + H nk where H n -k is (r& -fc)-square and 4-denotes direct sum.Of course, if r = k = n then (1.1) is the Hadamard determinant theorem: (1.3) Πhu^άetiH). 4 = 1 If some ha = 0, then H is singular and (1.3) is equality.If ha > 0, i = l, , n, then the condition (1.2) yields the well-known criterion for equality in (1.3), namely H = diag (fen, ••,/&««)-2* Results* Let /(a?) = /(ίCi, •••,«*) be a function defined for all nonnegative vectors x i> 0 (i.e., a? 0 if and only if x e C r .Also, / is strictly C r -monotone if f(x + u)> f(x), x e C r , u ^ 0, u Φ 0. THEOREM 1.Let H -(fe ίy ) be an n-square positive semi-definite 407