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 $B = A + \mathbf{U}\mathbf{U}^T$. For all $x \in \mathbb{R}$, we denote $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$. Let $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$. Show that the random variable $\chi_B(x)$ has finite expectation, and that, denoting by $\chi_A'$ the derivative of the polynomial $\chi_A$, we have $$\mathbb{E}\left[\chi_B(x)\right] = \chi_A(x) - \frac{1}{n} \chi_A'(x).$$
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 $B = A + \mathbf{U}\mathbf{U}^T$. For all $x \in \mathbb{R}$, we denote $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$. Let $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$. Show that the random variable $\chi_B(x)$ has finite expectation, and that, denoting by $\chi_A'$ the derivative of the polynomial $\chi_A$, we have
$$\mathbb{E}\left[\chi_B(x)\right] = \chi_A(x) - \frac{1}{n} \chi_A'(x).$$