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A proof of Sn-i (a11,..., ann) ≥ Sn-i (λ1,..., λn), 0 ≤ i ≤ n - 1 for an n × n positive definite symmetric matrix A = [aij]nxn, using mixed determinants
Author(s)
Pranayanuntana, Poramate
Date Issued
January 1, 2007
Type
Article
Abstract
By using mixed determinant of special form D (A, n - i; I, i) for positive definite symmetric matrix A, in particular the operator concavity property of the map f : A → D1/(n-i) (A, n - i; I, i)I together with unital positive linear map Φ : A → A ο I the Hadamard product of A with the identity matrix I, inequalities of the form Sn-i(a11,..., ann) ≥ Sn-i(λ1,..., λn), 0 ≤ i ≤ n - 1 for an n × n positive definite symmetric matrix A = [aij] with Sk : IRn → IR, k = 1, 2,...,n defined by Sk(χ) := ∑1≤i12<...< ik≤n χi1 χi2 · χik, and called the elementary symmetric polynomials and λi, i = 1, 2,...,n the eigenvalues of A, are derived. This result was first proved using Schur-concavity property of the elementary symmetric function Sk (χ) together with the fact that for any positive definite symmetric matrix A, the vector of its diagonal entries is majorized by the vector of its eigenvalues. Hence, this suggests relationship among mixed determinant, majorization and Schur-concavity.
Citation
Wseas Transactions on Mathematics, 6(1), 195-204, 2007