Let ${\bf A}$ be a size $n \times n$ symmetric positive semidefinite matrix with the first column being ${\bf a}_1$. If we define a new matrix, \begin{align} {\bf B} = \left[\begin{array}{cc} a_{11} & {\bf a}_1^T \\ {\bf a}_1 & {\bf A} \end{array}\right] \end{align}

How are the eigenvalues of ${\bf B}$ related to those of ${\bf A}$?

Let $\lambda_k$ and $\mu_k$ being the eigenvalues of ${\bf A}$ and ${\bf B}$, respectively. My simulations suggest that $\frac{\lambda_k}{n}$ and $\frac{\mu_k}{n+1}$ are very similar as $n \to \infty$. Can we prove this?