A subdeterminant inequality

Let A be an ^-square positive semi-definite hermitian matrix and let D m (A) denote the maximum of all order m principal subdeterminants of A. In this note we prove the inequality, m -1, , n -1 , and discuss in detail the case of equality.This result is closely related to Newton's and Sz£sz's inequalities.Let A -(an) be an ^-square positive semi-definite hermitian matrix with eigenvalues \ ^ Ξ> X n ;> 0. We introduce some notation.Fordenote the m-square submatrix of A whose (i,j) entry is a ω .ωj , i,j = l, , m. Q JNewton's inequalities [1, p. 106] state that if E m (A) is the mth elementary symmetric function of the nonnegative numbers λ x , , λ w then