MathDB

Putnam 2019 B3

Source:

December 10, 2019

Putnamlinear algebramatrixvectorPutnam 2019

Problem Statement

Let

Q

be an

n

-by-

n

real orthogonal matrix, and let

u\in \mathbb{R}^n

be a unit column vector (that is,

u^Tu=1

). Let

P=I-2uu^T

, where

I

is the

n

-by-

n

identity matrix. Show that if

1

is not an eigenvalue of

Q

, then

1

is an eigenvalue of

PQ

.

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