A proof of Σaj1···ajn-i≥ Σλj1···λjn-i, 0 ≤ ι ≤ n - 1 where ajk ∈ {a11, a22, ···, ann}, λjk∈ σ(A) for A = [aij] 0 IRn×n,using mixed determinants

  • Authors:

  • Poramate Pranayanuntana;Pattrawut Chansangiam

  • Affiliations:

  • King Mongkut's Institute of Technology Ladkrabang, Department of Control Engineering, Faculty of Engineering, Bangkok, Thailand;King Mongkut's Institute of Technology Ladkrabang, Department of Mathematics and Computer Science, Faculty of Science, Bangkok, Thailand

  • Venue:
  • MATH'06 Proceedings of the 10th WSEAS International Conference on APPLIED MATHEMATICS
  • Year:
  • 2006

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Abstract

We discuss the use of 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 º I the Hadamard product of A with the identity matrix I, in deriving an inequality of the form Σ aj1 ... ajn-i ≥ Σ λj1 ... λjn-i, 0 ≤ i ≤ n - 1, where the sums are taken over all (n-i)-tuples of positive integers (j1, ..., jn-i) whose entries do not exceed n, with λjk and ajk, 1 ≤ k ≤ n from the set of all n positive eigenvalues of A and the set of main diagonal entries of A, respectively.