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IMC 2014, Problem 7

Source: IMC 2014

July 27, 2016
IMCcollege contestslinear algebramatrixsuperior algebra

Problem Statement

Let A=(aij)i,j=1nA=(a_{ij})_{i, j=1}^n be a symmetric n×nn\times n matrix with real entries, and let λ1,λ2,…,λn\lambda _1, \lambda _2, \dots, \lambda _n denote its eigenvalues. Show that ∑1≤i<j≤naiiajj≥∑1≤i<j≤nλiλj\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j and determine all matrices for which equality holds.
(Proposed by Matrin Niepel, Comenius University, Bratislava)
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