Largest eigenvalue of a rank-one update

MediumLinear Algebra~4m

In $\mathbb{R}^4$ , let $v = (1,\,2,\,2,\,3)^{\top}$ and form the $4\times 4$ matrix $A = I + v v^{\top}$ (the identity plus the outer product of $v$ with itself). What is the largest eigenvalue of $A$ ?

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Don't expand the $4\times 4$ characteristic polynomial — recognize $vv^{\top}$ as rank-one.
$vv^{\top}$ has eigenvalues $\|v\|^2$ (once) and $0$ (thrice); adding $I$ shifts every eigenvalue up by 1.

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Asked at: Citadel, Two Sigma

Largest eigenvalue of a rank-one update

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