We now consider the case where $A \in \mathcal{S}_n(\mathbb{R})$ is symmetric. Let $\mathbf{u} \in \mathbb{R}^n$ be such that $\|\mathbf{u}\| = 1$. We set $B = A + \mathbf{u}\mathbf{u}^T$. Show that $B \in \mathcal{S}_n(\mathbb{R})$.
We now consider the case where $A \in \mathcal{S}_n(\mathbb{R})$ is symmetric. Let $\mathbf{u} \in \mathbb{R}^n$ be such that $\|\mathbf{u}\| = 1$. We set $B = A + \mathbf{u}\mathbf{u}^T$. Show that $B \in \mathcal{S}_n(\mathbb{R})$.