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Maximum and minimum problems in functions of quadratic forms Westwick, Roy

Abstract

Let A be an n x n hermitian matrix, let E₂(a₁, …, a_k) be the second elementary symmetric function of the letters a₁, …, a_k and let C₂(A) be the second compound matrix of A. In this thesis the maximum and minimum of det {(Ax_█, x_j)} and E₂ [(Ax₁, x₁), …, (Ax_█(k@), x_k)] the minimum of [formula omitted] (C₂(A)x_i ₁⋀x_i₂ , x_i₁ ⋀ax_i₂) are calculated. The maxima and minima are taken over all sets of k orthonormal vectors in unitary n-space and x_█(i@)₁ ⋀ x_i ₂ designates the Grassman exterior product. These results depend on the inequality E₂(a₁, …, a_k ) ≤ (k/2 ) [formula omitted] which is here established for arbitrary real numbers, and on the minimum of E₂ (x₁, …, x_(k)) where the minimum is taken over all values of x₁, …, x_█(k@) such that ∑_(i=1)^k▒xi = ∑_(i=1)^k▒〖∝i〗 and ∑_(i=1)^q▒xsi ≤ ∑_(i=1)^q▒〖∝i〗 for all sets of q distinct integers s₁, …, s_q taken from 1, …, k. Here α₁ ≥ … ≥ ∝_k.

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