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Suppose $x \in \mathbb{R}^n$ is a unit norm vector, and $\mathbf{U} \in \mathbb{R}^{n \times n}$ is a matrix such that $\|\mathbf{U}\|_F = 1$. We know that $x^T \mathbf{U} x = 0$.

From this fact, what can we say about $\|U x x^T \|_2$? Can we find an upper bound for that?

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