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


1 Answer 1


$$ \begin{pmatrix}1&0\\\0&k^{-1/2}\end{pmatrix} \begin{pmatrix}A_1&A_2\\\ kA_2^T&A_3\end{pmatrix} \begin{pmatrix}1&0\\\0&k^{1/2}\end{pmatrix} = \begin{pmatrix}A_1& k^{1/2}A_2\\\ k^{1/2}A_2^T&A_3\end{pmatrix} $$

Note that $A_3$ only needs to be symmetric.

  • $\begingroup$ What if $k$ is negative? Then there are obvious counterexamples even when all the $A_i$ are $1 \times 1$ matrices. $\endgroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.