0
$\begingroup$

Let $A$ be an $n\times n$ Hermitian matrix with eigenvalues $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. Prove that if $a_{ii} = \lambda_1$ for some $i$, then every other entry of row and column $i$ is zero. Simillarly prove that if $a_{ii} = \lambda_n$ for some $i$, then every other entry of row and column $i$ is zero.

Need some hints to solve the problem.

$\endgroup$
  • $\begingroup$ If you're thinking about Rayleigh-Ritz quotients, to what vectors $x$ do the numbers $a_{i,i}$ correspond as $x^{*}Ax/x^{*}x?$ $\endgroup$ – RideTheWavelet Sep 30 '17 at 3:35
  • $\begingroup$ Gershgorin's theorem might be of help.. $\endgroup$ – nemo Sep 30 '17 at 17:18

Your Answer

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

Browse other questions tagged or ask your own question.