In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.
In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$.
Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.