Q19
Matrices
Eigenvalue and Characteristic Polynomial Analysis
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We consider $A \in \mathcal{S}_n(\mathbb{R})$ symmetric with eigenvalues $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ and corresponding orthonormal basis of eigenvectors $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$. We set $B = A + \mathbf{u u}^T$ with $\|\mathbf{u}\| = 1$. Suppose in this question that $\lambda_1 < \lambda_2 < \cdots < \lambda_n$, and that $J = \{1,2,\ldots,n\}$. For $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$ we set $$f(x) = \sum_{k=1}^n \frac{\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle^2}{x - \lambda_k}.$$
(a) Show that $f$ is of class $C^\infty$ on $\mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, and calculate its derivative $f'(x)$.
(b) Show that the equation $f(x) = 1$ has a unique solution in each interval $]\lambda_\ell, \lambda_{\ell+1}[$ for all $\ell \in \{1,2,\ldots,n-1\}$, and in $]\lambda_n, +\infty[$.
(c) We denote by $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ the eigenvalues of $B$. Show that $$\lambda_1 < \mu_1 < \lambda_2 < \mu_2 < \cdots < \lambda_n < \mu_n.$$