IMC / 2013 / Problems / Day 1, P1

IMC 2013 · Day 1 · P1

Let $A$ and $B$ be real symmetric matrices with all eigenvalues strictly greater than 1. Let $\lambda$ be a real eigenvalue of matrix $AB$ . Prove that $|\lambda| > 1$ .

(Proposed by Pavel Kozhevnikov, MIPT, Moscow)

Solution (official)

The transforms given by $A$ and $B$ strictly increase the length of every nonzero vector, this can be seen easily in a basis where the matrix is diagonal with entries greater than 1 in the diagonal. Hence their product $AB$ also strictly increases the length of any nonzero vector, and therefore its real eigenvalues are all greater than 1 or less than $-1$ .

IMC 2013 · Day 1 · P1

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