In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Show that for all $k \in \{1,2,\ldots,n\}$, we have $$\mathbb{E}\left[\chi_B\left(\lambda_k\right)\right] = -\frac{1}{n} \chi_A'\left(\lambda_k\right).$$
In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Show that for all $k \in \{1,2,\ldots,n\}$, we have
$$\mathbb{E}\left[\chi_B\left(\lambda_k\right)\right] = -\frac{1}{n} \chi_A'\left(\lambda_k\right).$$