# Non symmetric matrices with real eigenvalues

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.

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$$\begin{pmatrix}1&0\\\0&k^{-1/2}\end{pmatrix} \begin{pmatrix}A_1&A_2\\\ kA_2^T&A_3\end{pmatrix} \begin{pmatrix}1&0\\\0&k^{1/2}\end{pmatrix} = \begin{pmatrix}A_1& k^{1/2}A_2\\\ k^{1/2}A_2^T&A_3\end{pmatrix}$$
Note that $A_3$ only needs to be symmetric.
What if $k$ is negative? Then there are obvious counterexamples even when all the $A_i$ are $1 \times 1$ matrices. –  Jeanne Clelland Mar 12 at 17:39
@Jeanne: The original post said all entries of $A$ are nonnegative, so $k\geq 0$ or $A_2 = 0$. –  Noah Stein Mar 12 at 18:12