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Let $G$ be a symmetric and indefinite matrix defined as follows

$$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$

where $S$ is a symmetric positive definite matrix of size $2n\times 2n$. Numerical result shows that $G$ is an indefinite matrix and its range of eigenvalues is $(a,b)\cup(c,d)$, where $a<b<0<c<d$.

I want to estimate the bounds $a,b,c,d$, but I have no idea now. I read some literatures and gain nearly nothing. Could anyone provide me any ideas or literatures about the estimate of eigenvalues of a symmetric and indefinite matrix?

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  • $\begingroup$ The first thing to try would probably be the circle theorem. Depending on the circumstance it can be really useful but sometimes it's not very informative without transforming the matrix somehow. $\endgroup$
    – Daniel Shapero
    Commented Oct 28, 2022 at 16:34
  • $\begingroup$ @RodrigodeAzevedo In fact, $S$ is a full matrix because $S$ is Schur complement of a sparse matrix $A$, and $A$ comes from the discretization of a second order elliptic equation. Sorry, I omit some information but I didn't expect to cause so much confusion. $\endgroup$
    – Nxy
    Commented Oct 29, 2022 at 13:52

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