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


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

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. $\endgroup$ – Jeanne Clelland Mar 12 '13 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 '13 at 18:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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