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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

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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.

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  • $\begingroup$ What if $k$ is negative? Then there are obvious counterexamples even when all the $A_i$ are $1 \times 1$ matrices. $\endgroup$
    – Jeanne Clelland
    Mar 12, 2013 at 17:39
  • $\begingroup$ @Jeanne: The original post said all entries of $A$ are nonnegative, so $k\geq 0$ or $A_2 = 0$. $\endgroup$
    – Noah Stein
    Mar 12, 2013 at 18:12

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