Consider the following block matrix


$A=\pmatrix{A\_1 & A\_2\cr kA\_2^\top & A\_3}$


where $A\_1$ is a symmetric matrix, $A\_3$ is diagonal matrix and all entries of $A$ are real and non-negative. 
How can we show that all eigenvalues of $A$ are real?


Note: $A\_2$ is not a square matrix.


Thanks in advance



Non symmetric matrices with real eigenvalues
$$
\begin{pmatrix}1&0\\\0&k^{-1/2}\end{pmatrix}
\begin{pmatrix}A\_1&A\_2\\\ kA\_2^T&A\_3\end{pmatrix}
 \begin{pmatrix}1&0\\\0&k^{1/2}\end{pmatrix}
 = \begin{pmatrix}A\_1& k^{1/2}A\_2\\\ k^{1/2}A\_2^T&A\_3\end{pmatrix}
$$


Note that $A\_3$ only needs to be symmetric.


