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B3

Part of 2019 Putnam

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Putnam 2019 B3

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12/10/2019
Let QQ be an nn-by-nn real orthogonal matrix, and let u∈Rnu\in \mathbb{R}^n be a unit column vector (that is, uTu=1u^Tu=1). Let P=I−2uuTP=I-2uu^T, where II is the nn-by-nn identity matrix. Show that if 11 is not an eigenvalue of QQ, then 11 is an eigenvalue of PQPQ.
Putnamlinear algebramatrixvectorPutnam 2019