How do get eigenvalues of a matrix B if add a row/column pair of a matrix A?

Question

I have a matrix of size N×N of the form:

where

and A is N-1 x N-1 matrix, a=0. I known the eigenvalues of A. Any possible for getting eigenvalues of B from eigenvalues of A?

Answer 1

If $Ax=\lambda x$, then it is easy to see that $$B\begin{bmatrix}\mu\\x\end{bmatrix} = \begin{bmatrix}a & C\\C^T & A\end{bmatrix}\begin{bmatrix}\mu\\x\end{bmatrix} = \begin{bmatrix}\mu a + s\\\lambda x\end{bmatrix}$$ where $s$ is the sum of the components of $x$. This means that if $\mu = \frac{s}{\lambda -a}$, you have $$B\begin{bmatrix}\mu\\x\end{bmatrix} = \begin{bmatrix}\mu a + s\\\lambda x\end{bmatrix} = \begin{bmatrix}a\frac{s}{\lambda - a} + s\\\lambda x\end{bmatrix}=\begin{bmatrix}\frac{as + \lambda s - as}{\lambda - a} \\\lambda x\end{bmatrix}=\\=\begin{bmatrix}\lambda \mu\\\lambda x\end{bmatrix}$$