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IMC 2013/1/1

Source: Imc 2013 Problem 1

August 8, 2013
linear algebramatrixvectorinequalitiesIMCcollege contests

Problem Statement

Let A\displaystyle{A} and B\displaystyle{B} be real symmetric matrixes with all eigenvalues strictly greater than 1\displaystyle{1}. Let λ\displaystyle{\lambda } be a real eigenvalue of matrix AB\displaystyle{{\rm A}{\rm B}}. Prove that ∣λ∣>1\displaystyle{\left| \lambda \right| > 1}.
Proposed by Pavel Kozhevnikov, MIPT, Moscow.
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