Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer 1

up vote 6 down vote accepted

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

Note that $A_3$ only needs to be symmetric.

share|improve this answer
    
perfect! thank you so much –  Jamil Tau Mar 12 '13 at 13:10
    
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 '13 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 '13 at 18:12
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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